<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

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Classroom Tips and Techniques: Tensor Calculus with the Differential Geometry Package

Robert J. Lopez

Emeritus Professor of Mathematics and Maple Fellow

Maplesoft

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<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

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The Tensor subpackage of the DifferentialGeometry package supplants the now-deprecated tensor package in Maple. The tensor package made essential use of the also-deprecated linalg package, so although worksheets that used these deprecated packages still work, it is imperative to move to using the new formalisms in the Tensor package.

This article is a "survivor's guide" for implementing tensor calculus in the new Tensor package. It explains the constructs in this package from the perspective of classical (i.e., indicial) tensor notation.

The DifferentialGeometry package itself contains some 34 command but also six subpackages, themselves contributing to a total of some 184 commands. These subpackages, and a measure of their "size" are listed in Table 1.

Subpackage

Number of Commands

Tensor

Tools

JetCalculus

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEjMjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGLw==

GroupActions

Library

LieAlgebras

Table 1 Subpackages in the DifferentialGeometry package

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The complexity of mastering the DifferentialGeometry package is further increased when subpackages such as GroupActions themselves have subpackages (MovingFrames). A comprehensive tutorial in the complete DifferentialGeometry package would require more than a textbook, so clearly, this is not our ambition here. Fortunately, the package itself contains two sets of useful tutorials, a comprehensive collection of Lessons, and a set of Tutorials. The Lessons worksheets provide a systematic approach to learning the commands in the DifferentialGeometry, Tensor, LieAlgebras and JetCalculus subpackages. Each lesson also contains a set of exercises that range in difficulty from simple computations to programming problems. Solutions are given. The tutorials present specialized applications of the DifferentialGeometry package.

Our more modest goal for this article is to show how to enter covariant and contravariant tensors, compute their covariant derivatives, obtain the equations of parallel transport and geodesics, and compute the basic tensors of general relativity.

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<Text-field style="Heading 1" layout="Heading 1">Initializations</Text-field>

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<Text-field style="Heading 1" layout="Heading 1">Declaring the Frame in DifferentialGeometry</Text-field>

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Before a vector or tensor can be entered, a frame must be declared by stating its variables and giving it a name. For example, to declare LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 as the Cartesian space with variables LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, execute

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

SS9mcmFtZX5uYW1lOn5SMkc2Ig==

The default behavior for DifferentialGeometry is that from this point onward, the prompt would be modified to display the frame name for as long as that frame were the active one. Thus, without our having suspended this default with the Preferences command in the Initializations section, the prompt following the DGsetup command would be the one shown in Table 2.

Table 2 The default modification of the prompt

We have elected to suspend this default behavior for three reasons. First, interactive editing of a worksheet can create a confusing display. If in a section where one frame name appears in all prompts, a new frame is defined, the subsequent prompts that show the earlier name do not change to show the new name. Only a "new" prompt will display the new frame name. The result is a worksheet with prompts showing different frame names where they might not be relevant. Such misplaced prompts have to be deleted manually if they are not to provide incorrect information.

Second, these modified prompts are persistent - they cannot be removed by any Maple command. They have to be removed by deletion. (The Maple interface command that modifies prompts does not cascade the change through existing prompts. It only modifies new prompts.)

Finally, if the commands are executed in Document Blocks, and not at prompts, there will be no visible prompt to modify. Thus, the modified prompt is a worksheet paradigm that does not carry through all of Maple usage. For all these reasons, we will not have frame names visible in our prompts.

JSFH

<Text-field style="Heading 1" layout="Heading 1">The Contravariant/Covariant Paradigms</Text-field>

JSFH

Anyone who had taken even a perfunctory dip into the waters of tensor calculus knows there are two words, covariant and contravariant, that must be faced. We will not be able to enter a tensor in the Tensor package without making the distinction between these two terms. Using the Einstein summation convention (repeated indices, one raised and one lowered, are summed), Table 3 defines contravariant and covariant vectors.

Vector Type

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiVkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjcvRjZRJ25vcm1hbEYn

Basis Type

Transformation Law

Contravariant

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JJW1zdWJHRiQ2JS1GLzYnUSJlRicvJSVib2xkR0Y0L0YzUSZmYWxzZUYnL0Y2USVib2xkRicvJStmb250d2VpZ2h0R0ZLRjgvJS9zdWJzY3JpcHRzaGlmdEdGPy9GNlEnbm9ybWFsRic=

Tangents to coordinate curves

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

Covariant

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSVtc3VwR0YkNiUtRi82J1EiZUYnLyUlYm9sZEdGNC9GM1EmZmFsc2VGJy9GNlElYm9sZEYnLyUrZm9udHdlaWdodEdGS0Y4LyUxc3VwZXJzY3JpcHRzaGlmdEdGPy9GNlEnbm9ybWFsRic=

Gradients (Normal to coordinate surfaces)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRImtGJ0YyRjUtSSNtb0dGJDYtUSInRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnRjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUY+Ni1RIj1GJ0ZBRkNGRkZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGaW4tRiw2JUYuLUYjNiUtRi82JVEiaUYnRjJGNUYyRjVGWC1JJm1mcmFjR0YkNigtRiM2Ji1GPjYtUScmcGFydDtGJ0ZBRkNGRkZIRkpGTEZORlAvRlNGVy9GVkZULUklbXN1cEdGJDYlLUYvNiVRInhGJ0YyRjVGXW8vJTFzdXBlcnNjcmlwdHNoaWZ0R0ZaLUY+Ni1RIn5GJ0ZBRkNGRkZIRkpGTEZORlBGam9GVUZBLUYjNiZGZ28tRl1wNiUtRi82JVEidUYnRjJGNUY4RmJwRjJGNS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGY3EvJSliZXZlbGxlZEdGRUZB

Table 3 Contravariant and covariant vectors

In the rightmost column of Table 3 uses the notation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn for components expressed in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-coordinate system, but LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRImtGJ0YyRjUtSSNtb0dGJDYtUSInRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnRjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRImtGJ0YyRjUtSSNtb0dGJDYtUSInRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnRjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkE= for components in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-coordinate system. Texts also denote the new coordinate system bu the use of an overbar on the component, or a prime on the left side of the variable.

The components LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn are the contravariant and covariant components, respectively, of the vector V. The basis vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjlRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GOVEnbm9ybWFsRic= are reciprocal, so that 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. Thus, an orthonormal basis is self-reciprocal. That is why the distinction between contravariant and covariant basis does not matter in Cartesian spaces.

If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MzQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is the mapping from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJy9GNlEnbm9ybWFsRic= via functions of the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EidUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYvUSI9RidGL0Y1RjgvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZORistSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYnUSJ4RidGL0YyRjVGOC9GNlEnbm9ybWFsRidGWUZZ, then the gradient vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GOVEnbm9ybWFsRic= are the rows of the Jacobian matrix 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, where the upper index LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is interpreted as a row index, and the lower index LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, as a column index.

If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5MzQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYkUSIxRidGNS9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNQ== is the mapping from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= via functions of the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYvUSI9RidGL0Y1RjgvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZORistSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYnUSJ1RidGL0YyRjVGOC9GNlEnbm9ybWFsRidGWUZZ, then the tangent vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjlRJ25vcm1hbEYn are the columns of the Jacobian matrix 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.

JSFH

To facilitate the implementation of the contravariant transformation law, writing the components LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= as a column vector v means the sums with the Jacobian matrix are along a row and across the columns of the matrix. Hence, the matrix product LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiQtSSVtc3ViR0YkNiUtSSNtaUdGJDYnUSJKRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPy1GIzYlLUY0NiVRJyYjOTM0O0YnRjovRj5RJ25vcm1hbEYnL0Y7RjkvRj5RJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGR0ZHLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUY0NidRInZGJ0Y3RjpGPUZARkc= implements the contravariant transformation. Writing the components LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn as the row vector w means the sums with the Jacobian matrix are down a column but across the rows. Hence, the matrix product 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 implements the covariant transformation. This inherent distinction between tangent bases and normal bases induces the distinction between contravariant and covariant. Using column and row vectors to express this difference is a convenient visual device in classical tensor calculus.

<Text-field style="Heading 1" layout="Heading 1">The Tensor as a Multilinear Object</Text-field>

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If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a vector space, say, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 with basis 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, then a rank-two tensor is a multilinear object LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RLCZiaWdvdGltZXM7RicvJSVzaXplR1EiOEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZPRitGPA==, the direct product of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with itself, having doubly-indexed basis objects LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiQtSSVtc3ViR0YkNiUtSSNtaUdGJDYnUSJlRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPy1GIzYlLUY0NiVRI2lqRicvRjtGOS9GPlEnaXRhbGljRidGR0ZILyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRj5RJ25vcm1hbEYnRk0vJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRk0=. The tensor is actually the object

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linear in both indices. Of course, the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn are the contravariant components of the tensor; and just as for vectors, there would be the equivalent covariant components, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==. There are even mixed tensors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImpGJ0YyRjVGMkY1LUYjNiUtRi82JVEiaUYnRjJGNUYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkQvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnL0Y2USdub3JtYWxGJw== that transform contravariantly in one index but covariantly in another.

In actual practice, one manipulates just the components of the tensor, and almost never explicitly exhibits the basis objects. However, in the DifferentialGeometry package, vectors and tensors require an explicit use of the basis objects.

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<Text-field style="Heading 1" layout="Heading 1">Bases and Their Duals</Text-field>

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The basis 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1 could be entered as

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

NyQtSSRfREdHNiI2IzckNyVRJ3ZlY3RvckYlSSNSMkdGJTciNyM3JDcjIiIiRi8tRiQ2IzckRig3IzckNyMiIiNGLw==

or could be extracted from Maple with the DGinfo command from the Tools package.

NyQtSSRfREdHNiI2IzckNyVRJ3ZlY3RvckYlSSNSMkdGJTciNyM3JDcjIiIiRi8tRiQ2IzckRig3IzckNyMiIiNGLw==

(To type the underscore in math (2D) mode, press the escape character (\134) first. Alternatively, enter such expressions in text (linear, 1D) mode and convert to math mode via the Context Menu.)

The reciprocal (or dual) basis 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 is then

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

NyQtSSRfREdHNiI2IzckNyVRJWZvcm1GJUkjUjJHRiUiIiI3IzckNyNGK0YrLUYkNiM3JEYoNyM3JDcjIiIjRis=

In actual fact, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is considered a differential form, more in keeping with the modern approach to differential geometry.

JSFH

<Text-field style="Heading 1" layout="Heading 1">Representing Vectors as DifferentialGeometry Objects</Text-field>

JSFH

In the DifferentialGeometry package, the contravariant vector whose components are 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 is given by

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiYtRiw2JVEmdl9jb25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNiUtRiw2JVEnZXZhbERHRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYlRistRiM2J0YrLUYjNiZGKy1GIzYlLUYsNiVRIkFGJ0Y2RjlGZW4tRlxvNiQtRiM2JS1GLDYlUSJ4RidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GaG4vRlVRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRjZGOUZALUY9Ni1RMSZJbnZpc2libGVUaW1lcztGJ0ZARkJGRUZHRklGS0ZNRk9GaG5Gam4tRiw2JVEkRF94RidGNkY5LUY9Ni1RIitGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGY3EtRiM2JkYrLUYjNiUtRiw2JVEiQkYnRjZGOUZlbkZpb0ZpcC1GLDYlUSREX3lGJ0Y2RjlGK0YrRkBGK0YrRis=

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkjUjJHRiQ3IjckNyQ3IyIiIi1JIkFHRiQ2JEkieEclKnByb3RlY3RlZEdJInlHRjM3JDcjIiIjLUkiQkdGJEYx

or by

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkjUjJHRiQ3IjckNyQ3IyIiIi1JIkFHRiQ2JEkieEclKnByb3RlY3RlZEdJInlHRjM3JDcjIiIjLUkiQkdGJEYx

The evalDG and the DGzip commands are two of the simpler ways to create an object whose data structure is intrinsic to the DifferentialGeometry package. When using the evalDG command, the asterisk is the explicit multiplication operator. If this vector had been entered in math mode, the echo of the asterisk would be a centered dot. The alternative to the asterisk would be the space.

The covariant vector whose components are 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 is entered as

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

LUkkX0RHRzYiNiM3JDclUSVmb3JtRiRJI1IyR0YkIiIiNyQ3JDcjRiotSSJhR0YkNiRJInhHJSpwcm90ZWN0ZWRHSSJ5R0YyNyQ3IyIiIy1JImJHRiRGMA==

or as

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

LUkkX0RHRzYiNiM3JDclUSVmb3JtRiRJI1IyR0YkIiIiNyQ3JDcjRiotSSJhR0YkNiRJInhHJSpwcm90ZWN0ZWRHSSJ5R0YyNyQ3IyIiIy1JImJHRiRGMA==

There does not seem to be a simple way to represent a vector as a column or row vector. The Tools subpackage provides the DGinfo command with which the components of a vector can be extracted. Its use is illustrated by

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

NyQ3JDcjIiIiLUkiQUc2IjYkSSJ4RyUqcHJvdGVjdGVkR0kieUdGKzckNyMiIiMtSSJCR0YoRik=

or by

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEnREdpbmZvRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSZ2X2NvbkYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRictSSNtc0dGJDYjUTBDb2VmZmljaWVudExpc3RGJ0Zobi1GVzYmLUYjNiUtRlc2Ji1GIzYjLUkjbW5HRiQ2JFEiMUYnRkBGQC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zobi1GVzYmLUYjNiMtRltwNiRRIjJGJ0ZARkBGXnBGYXBGQEZecEZhcEZARitGKw==

NyQtSSJBRzYiNiRJInhHJSpwcm90ZWN0ZWRHSSJ5R0YoLUkiQkdGJUYm

<Text-field style="Heading 1" layout="Heading 1">Representing Tensors as DifferentialGeometry Objects</Text-field>

JSFH

A rank-two contravariant tensor would be entered as

JSFH

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkjUjJHRiQ3JDckUShjb25fYmFzRiRGLDciNyY3JDckIiIiRjFJJFQxMUdGJDckNyRGMSIiI0kkVDEyR0YkNyQ3JEY1RjFJJFQyMUdGJDckNyRGNUY1SSRUMjJHRiQ=

The construct LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEkRF94RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIiZGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVRInRGJ0YvRjJGNUYrRjk= corresponds to the dyadic basis element LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1JI21uR0YkNiRRIjFGJy9GOVEnbm9ybWFsRicvRjZGNC9GOVEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrRkM=, etc.

A rank-two covariant tensor would be entered as

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkjUjJHRiQ3JDckUShjb3ZfYmFzRiRGLDciNyY3JDckIiIiRjFJJHQxMUdGJDckNyRGMSIiI0kkdDEyR0YkNyQ3JEY1RjFJJHQyMUdGJDckNyRGNUY1SSR0MjJHRiQ=

The construct LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEjZHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEiJkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVEidEYnRi9GMkY1RitGOQ== corresponds to the dyadic basis element LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1JI21uR0YkNiRRIjFGJy9GOVEnbm9ybWFsRicvRjZGNC9GOVEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGQw==, etc.

The components of a tensor can be recovered with the DGinfo command

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEnREdpbmZvRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSdUMl9jb25GJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUkjbXNHRiQ2I1ExT2JqZWN0Q29tcG9uZW50c0YnRitGQEYrRis=

NyY3JDckIiIiRiVJJFQxMUc2IjckNyRGJSIiI0kkVDEyR0YnNyQ3JEYqRiVJJFQyMUdGJzckNyRGKkYqSSRUMjJHRic=

or with the convert command.

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

The components of the rank-two tensor are often represented as the entries of a matrix. If a matrix is used for such a representation, it is possible to convert the matrix to a DifferentialGeometry tensor.

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkjUjJHRiQ3JDckUShjb25fYmFzRiRGLDciNyY3JDckIiIiRjFJJFQxMUdGJDckNyRGMSIiI0kkVDEyR0YkNyQ3JEY1RjFJJFQyMUdGJDckNyRGNUY1SSRUMjJHRiQ=

<Text-field style="Heading 1" layout="Heading 1">Contraction of Indices</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Given two tensors of conformable dimensions, say LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lrRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2prRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==, forming the sum of products LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lrRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSVtc3ViR0YkNiUtRi82JVEiVkYnRjJGNS1GIzYlLUYvNiVRI2prRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R0Y/L0Y2USdub3JtYWxGJw== is the operation contraction of indices. Recall that the Einstein summation convention indicates a sum on the same index when it appears once raised and once lowered. This operation, implemented in the Tensor package with the command ContractIndices, is illustrated for the two tensors

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnVDJfY292RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUSdUMl9jb25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGKw==

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkjUjJHRiQ3JDckUShjb3ZfYmFzRiRGLDciNyY3JDckIiIiRjFJJHQxMUdGJDckNyRGMSIiI0kkdDEyR0YkNyQ3JEY1RjFJJHQyMUdGJDckNyRGNUY1SSR0MjJHRiQ=

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkjUjJHRiQ3JDckUShjb25fYmFzRiRGLDciNyY3JDckIiIiRjFJJFQxMUdGJDckNyRGMSIiI0kkVDEyR0YkNyQ3JEY1RjFJJFQyMUdGJDckNyRGNUY1SSRUMjJHRiQ=

from the previous section. To simplify data entry, we rename the first as T and the second, as V.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiw2JVEnVDJfY292RidGNkY5RistRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiw2JVEnVDJfY29uRidGNkY5RistRj02LVEiOkYnRkBGQkZFRkdGSUZLRk1GT0ZRRlQ=

The four possible contractions that result in a rank-two tensor are given in Table 4, where their components are displayed as elements of matrices.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKmNzLV8i

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

Table 4 Four rank-two tensors formed by contraction of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2ttRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Careful inspection of the four tensors in Table 4 (see especially the rightmost column) shows that they are all different.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">The Metric Tensor</Text-field>

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The geometry of a manifold is first captured in the covariant metric tensor 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 or its contravariant counterpart 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. There is no "calculus" in tensor calculus without first obtaining this essential tensor. Hence, it is imperative that there be efficient ways to obtain this tensor. Several of these techniques will be illustrated for the Cartesian plane on which polar coordinates have been imposed.

<Text-field style="Heading 2" layout="Heading 2">Method 1 - Obtain <Equation executable="false" style="Heading 2" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZ0YnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEjaWpGJ0YyRjVGOEYyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0YyL0Y5USdub3JtYWxGJw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZ0YnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEjaWpGJ0YyRjVGOEYyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0YyL0Y5USdub3JtYWxGJw==</Equation> as a Matrix and Convert to a Tensor</Text-field>

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Define the map LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5MzQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYkUSIxRidGNS9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNQ== with equations of the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUEYrLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2J1EidUYnRi9GMkY1RjhGPkY+Rj4= via

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

KiZJInJHNiIiIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kmdGhldGFHRiRGJQ==

KiZJInJHNiIiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kmdGhldGFHRiRGJQ==

and the radius (position) vector via

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

Then, a representation of the basis vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSVtc3ViR0YkNiUtSSNtaUdGJDYnUSJlRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y+USdub3JtYWxGJy9GO0Y5L0Y+USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRkgvJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlRjMtRiM2JS1GRTYkUSIyRidGSEZKRktGTUZIRkgvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRkg= is given by

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdRJmZhbHNlRidGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEjOj1GJ0Y+LyUmZmVuY2VHRkUvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGZ24tSShtYWN0aW9uR0YkNiUtSShtZmVuY2VkR0YkNigtRiM2Ji1JJ210YWJsZUdGJDY2LUkkbXRyR0YkNiYtSSRtdGRHRiQ2KC1GIzYoLUYvNiVRJGNvc0YnL0YzRkVGPi1GSjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+Rk1GT0ZRRlNGVUZXRlkvRmZuUSYwLjBlbUYnL0ZpbkZlcC1GXm82JC1GIzYmLUYvNiVRJyYjOTUyO0YnRmBwRj5GQEZDRj5GPkZARkNGPi8lKXJvd2FsaWduR1EhRicvJSxjb2x1bW5hbGlnbkdGYHEvJStncm91cGFsaWduR0ZgcS8lKHJvd3NwYW5HRj0vJStjb2x1bW5zcGFuR0Y9Rl5xRmFxRmNxLUZmbzYmLUZpbzYoLUYjNigtRi82JVEkc2luRidGYHBGPkZhcEZncEZARkNGPkZecUZhcUZjcUZlcUZncUZecUZhcUZjcS8lJmFsaWduR1ElYXhpc0YnL0ZfcVEpYmFzZWxpbmVGJy9GYnFRJ2NlbnRlckYnL0ZkcVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjQvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0Zfcy8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0Zqcy8lJmZyYW1lR0Zqcy8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRkUvJS1lcXVhbGNvbHVtbnNHRkUvJS1kaXNwbGF5c3R5bGVHRkUvJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGZ3NGQEZDRj5GPi9JK21zZW1hbnRpY3NHRiRRKkNvbFZlY3RvckYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRl11LyUrYWN0aW9udHlwZUdRLnJ0YWJsZWFkZHJlc3NGJy8lKXJ0YWJsZWlkR1EqMTUyMTQ1ODA4RidGQEZDRj4=

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

so that a matrix whose entries are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw== is

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

To convert this to the metric tensor in polar coordinates, we need to define the polar frame with

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

in which case we then have

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyRRKGNvdl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpSSJyRyUqcHJvdGVjdGVkR0Y0RjE=

The covariant form of this metric tensor, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, can be obtained with

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyRRKGNvbl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpSSJyRyUqcHJvdGVjdGVkR0Y0ISIi

The matrix representing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is the inverse of the matrix representing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==.

<Text-field style="Heading 2" layout="Heading 2">Method 2 - Transform the Cartesian Metric Tensor</Text-field>

JSFH

Begin with the contravariant metric tensor on the Cartesian space LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1:

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkjUjJHRiQ3JDckUShjb3ZfYmFzRiRGLDciNyQ3JDckIiIiRjFGMTckNyQiIiNGNEYx

Define the transformation from Cartesian to polar coordinates as

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

LUkkX0RHRzYiNiM3JDcmUS90cmFuc2Zvcm1hdGlvbkYkNyQ3JEkiUEdGJCIiITckSSNSMkdGJEYsNyI3Iy1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjL0kkJWlkR0YkIipja2R4IjckNyQqJkkickdGNCIiIi1JJGNvc0dGMzYjSSZ0aGV0YUdGNEY+SSJ4R0Y0NyQqJkY9Rj4tSSRzaW5HRjNGQUY+SSJ5R0Y0

and use it to convert the Euclidean metric to

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEpUHVsbGJhY2tGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKFBoaV9pbnZGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRI2dFRidGNkY5RkBGK0Yr

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyRRKGNvdl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpSSJyRyUqcHJvdGVjdGVkR0Y0RjE=

<Text-field style="Heading 1" layout="Heading 1">Raising and Lowering Indices</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The same vector V can be given with respect to the natural tangent basis vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjlRJ25vcm1hbEYn or with respect to the reciprocal basis of gradient basis vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GOVEnbm9ybWFsRic=. Thus, one has 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. The conversion between contravariant and covariant components of V is effected by contraction with the metric tensor:

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 or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVy1JJW1zdWJHRiQ2JUYuLUYjNiUtRi82JVEiaUYnRjJGNUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPy1GQTYtUSJ+RidGREZGRklGS0ZNRk9GUUZTL0ZWUSYwLjBlbUYnL0ZZRmJvLUYsNiUtRi82JVEiZ0YnRjJGNS1GIzYlLUYvNiVRI2lrRidGMkY1RjJGNUY9RkQ=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Example 1</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Given the contravariant vector

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiLUkiQUdGJDYkSSJyRyUqcHJvdGVjdGVkR0kmdGhldGFHRjM3JDcjIiIjLUkiQkdGJEYx

the covariant components are given by the RaiseLowerIndices command in the Tensor package. Thus, we have

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyNRKGNvdl9iYXNGJDciNyQ3JDcjIiIiLUkiQUdGJDYkSSJyRyUqcHJvdGVjdGVkR0kmdGhldGFHRjY3JDcjIiIjKiYpRjVGOkYxLUkiQkdGJEY0RjE=

where "g" denotes the covariant metric tensor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==. Alternatively, given the covariant vector

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

LUkkX0RHRzYiNiM3JDclUSVmb3JtRiRJIlBHRiQiIiI3JDckNyNGKi1JImFHRiQ2JEkickclKnByb3RlY3RlZEdJJnRoZXRhR0YyNyQ3IyIiIy1JImJHRiRGMA==

the contravariant components are

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyNRKGNvbl9iYXNGJDciNyQ3JDcjIiIiLUkiYUdGJDYkSSJyRyUqcHJvdGVjdGVkR0kmdGhldGFHRjY3JDcjIiIjKiYpRjVGOiEiIi1JImJHRiRGNEYx

where "LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=" denotes the contravariant metric tensor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">The Connection Coefficients</Text-field>

JSFH

Once the metric tensor is known, the way the basis vectors change from point to point can be determined. It turns out that the rate of change of the basis vectors can be expressed as linear combinations of these same vectors. The coefficients of these linear combinations are called the connection coefficients, or the Christoffel symbols. Depending on the form used for these symbols, they are called the Christoffel symbols of the first kind or the second kind. Since there is a built-in command that provides the Christoffel symbols of the second kind, we will obtain those via

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomY29sb25lcTtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GU0Y1LUYsNiVRLENocmlzdG9mZmVsRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEiZ0YnRi9GMkY5LUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkhGSkZU

LUkkX0RHRzYiNiM3JDclUStjb25uZWN0aW9uRiRJIlBHRiQ3JDclUShjb3ZfYmFzRiRRKGNvbl9iYXNGJEYsNyI3JTckNyUiIiIiIiNGMyokSSJyRyUqcHJvdGVjdGVkRyEiIjckNyVGM0YyRjMsJEY1Rjc3JDclRjNGM0YyRjQ=

Classical texts in differential geometry use the notation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEnJiM5MTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRI2prRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LUYjNiUtRi82JVEiaUYnRj1GP0Y9Rj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkgvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnRjU=, 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, or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSNqa0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiaUYnRjRGN0Y+Rj4vJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRj4= for Christoffel symbols of the second kind, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSNqa0YnRjRGN0Y+Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGPg== or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSNqa0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiaUYnRjRGN0Y+Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGPg== for Christoffel symbols of the first kind. Consequently, the correct interpretation of the output of the Christoffel command is captured, painstakingly and laboriously, in Table 5.

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

Table 5 Christoffel symbols of the second kind for polar coordinates

The closest the Tensor package comes to articulating the Christoffel symbols is

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnREdpbmZvRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEiQ0YnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictSSNtc0dGJDYjUTFPYmplY3RDb21wb25lbnRzRictRiw2I1EhRidGQQ==

NyU3JDclIiIiIiIjRiYqJEkickclKnByb3RlY3RlZEchIiI3JDclRiZGJUYmLCRGKEYqNyQ3JUYmRiZGJUYn

where the list LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNigtSSNtaUdGJDYlUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJqRidGNEY3RjotRjE2JVEia0YnRjRGN0Y+Rj4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGPg== maps to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEnJiM5MTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRI2lrRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LUYjNiUtRi82JVEiakYnRj1GP0Y9Rj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkgvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnRjU=. Thus, the middle index in the list is the raised one, and the first and third are the lower ones.

<Text-field style="Heading 2" layout="Heading 2">Example 2</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

For polar coordinates, show that

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

Thus, show that the derivative of a basis vector can be expressed as a linear combination of the basis vectors, and that the coefficients of these linear combinations are the Christoffel symbols.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

To show that derivatives of basis vectors can be expressed as linear combinations of basis vectors, begin by expressing 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 in terms of 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. This is done in Table 6 where the symbols i and j are introduced explicitly and naming clashes are avoided by introducing the alternate names 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 for 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.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCQvSSJpRzYiLCQqJiwmKiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmdGhldGFHRi0iIiImSSJlR0YlRi9GMUYxKihJInJHRi1GMSZGMzYjRjVGMS1JJGNvc0dGLEYvRjEhIiJGMUY1RjpGOi9JImpHRiUqJkY1RjosJiooRjVGMUYqRjFGNkYxRjEqJkYyRjFGOEYxRjFGMQ==

Table 6 Expressing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYnUSJpRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPC1JI21vR0YkNi1RIixGJy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYnUSJqRidGNEY3RjpGPUZDRkMvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRkM= in terms of 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Table 7, which lists the derivatives of the basis vectors, shows that the Christoffel symbols indeed are the coefficients in the linear combinations of basis vectors that express the derivatives of the basis vectors.

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqJkkickclKnByb3RlY3RlZEciIiImSSJlRzYiNiNGJEYmISIi

Table 7 Differentiation of basis vectors in polar coordinates

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Table 8 provides a formula for computing Christoffel symbols of the second kind from the components of the metric tensor.

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Table 8 Christoffel symbols in terms of the metric tensor

JSFH

A practical notational shortcut is the use of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbW9HRiQ2LVEnJnBhcnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjExMTExMTFlbUYnLUYjNiUtSSNtaUdGJDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnRlBGUy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjI= for the differentiation operator 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. The formula in Table 8 would be easier to write (and even remember) if this notational device is used.

<Text-field style="Heading 1" layout="Heading 1">The Covariant Derivative</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

In calculus, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYnUSJ1RicvJSVib2xkR1EldHJ1ZUYnRjIvRjZRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkEvRjNGPy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYvNiVRImZGJ0ZERkVGNUY1LUYvNiNRIUYnRjU= is the directional derivative of the scalar function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= taken in the direction of the unit vector u. The gradient vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEmJkRlbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8= arises naturally from the calculation of the derivative

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

which is how the directional derivative is defined when 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, 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, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2J1EidUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1JJW1zdWJHRiQ2JS1GLDYlRi4vRjNGMS9GNlEnaXRhbGljRictRiM2JS1JI21uR0YkNiRRIjFGJ0Y+RlhGWS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUY7Ni1RIn5GJ0Y+RkBGQkZERkZGSEZKRkwvRk9RJjAuMGVtRicvRlJGYm8tRiw2J1EiaUYnRi9GMkY1RjgtRjs2LVEiK0YnRj5GQEZCRkRGRkZIRkpGTC9GT1EsMC4yMjIyMjIyZW1GJy9GUkZbcC1GVDYlRlYtRiM2JS1GaG42JFEiMkYnRj5GWEZZRltvRl5vLUYsNidRImpGJ0YvRjJGNUY4Rj4=. Of course, this calculation extends to higher dimensions. But more important, note how starting with a scalar function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the vector quantity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEmJkRlbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8= must be defined, and the desired directional derivative is a dot product of this gradient vector with the direction vector.

The covariant derivative arises in much the same way, that is, from defining a directional derivative of a vector. The new object that must be created is the covariant derivative, a rank-two tensor, and the actual rate of change of the vector in a given direction is the "dot product" of this new tensor with a vector specifying the direction. In particular, Table 9 gives the expressions for the covariant derivative of contravariant and covariant vectors, a mixed tensor, and the metric tensor.

Contravariant vector

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

Covariant vector

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

Mixed tensor

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

Metric tensor

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

Table 9 Formulas for covariant derivatives

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

In polar coordinates, covariant derivatives of the contravariant and covariant vectors

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEncHZfY29uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUSdwdl9jb3ZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGKw==

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiLUkiQUdGJDYkSSJyRyUqcHJvdGVjdGVkR0kmdGhldGFHRjM3JDcjIiIjLUkiQkdGJEYx

LUkkX0RHRzYiNiM3JDclUSVmb3JtRiRJIlBHRiQiIiI3JDckNyNGKi1JImFHRiQ2JEkickclKnByb3RlY3RlZEdJJnRoZXRhR0YyNyQ3IyIiIy1JImJHRiRGMA==

respectively, are obtained in the Tensor package with the CovariantDerivative command. It requires as a second argument the connection coefficients, which can be given either as the Christoffel symbols (of the second kind)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

LUkkX0RHRzYiNiM3JDclUStjb25uZWN0aW9uRiRJIlBHRiQ3JDclUShjb3ZfYmFzRiRRKGNvbl9iYXNGJEYsNyI3JTckNyUiIiIiIiNGMyokSSJyRyUqcHJvdGVjdGVkRyEiIjckNyVGM0YyRjMsJEY1Rjc3JDclRjNGM0YyRjQ=

or as the connection

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVErQ29ubmVjdGlvbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEiQ0YnRjZGOUZARitGKw==

LUkkX0RHRzYiNiM3JDclUStjb25uZWN0aW9uRiRJIlBHRiQ3JDclUShjb3ZfYmFzRiRRKGNvbl9iYXNGJEYsNyI3JTckNyUiIiIiIiNGMyokSSJyRyUqcHJvdGVjdGVkRyEiIjckNyVGM0YyRjMsJEY1Rjc3JDclRjNGM0YyRjQ=

(In the Tensor package, the Christoffel symbols are not separated. The argument to the Connection command is the same sum of terms that the Tensor package uses to express the Christoffel symbols.)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The covariant derivative of the contravariant vector is

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyRRKGNvbl9iYXNGJFEoY292X2Jhc0YkNyI3JjckNyQiIiJGMi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJBR0YkNiRJInJHRjVJJnRoZXRhR0Y1Rjo3JDckRjIiIiMsJiomRjpGMi1JIkJHRiRGOUYyISIiLUY0NiRGN0Y7RjI3JDckRj5GMiomLCZGQUYyKiYtRjQ2JEZBRjpGMkY6RjJGMkYyRjpGQzckNyRGPkY+KiYsJkY3RjIqJi1GNDYkRkFGO0YyRjpGMkYyRjJGOkZD

more easily read as the array

LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKk9tPl8i

<Text-field style="Heading 1" layout="Heading 1">The Directional Covariant Derivative</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The covariant derivative 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 is the rank-two tensor that arises when the directional derivative of the vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= is defined. As such, it is the generalization of the gradient vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEmJkRlbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictSSNtaUdGJDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8= that arises when the directional derivative of the scalar function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is defined. Just as the directional derivative of the scalar function requires a dot product with a direction vector, so too does the directional derivative of the vector. Thus, the directional derivative of the vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= in the direction LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRImpGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjU= is given by 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, the contraction of the covariant derivative with the direction vector.

This calculation is implemented in the Tensor package via the command DirectionalCovariantDerivative.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Example 3</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The derivative of the contravariant vector

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUSdwdl9jb25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGKw==

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiLUkiQUdGJDYkSSJyRyUqcHJvdGVjdGVkR0kmdGhldGFHRjM3JDcjIiIjLUkiQkdGJEYx

in the direction of the vector

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiSSJ1R0YkNyQ3IyIiI0kidkdGJA==

is given by

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyNRKGNvbl9iYXNGJDciNyQ3JDcjIiIiLCgqKEkickclKnByb3RlY3RlZEdGMUkidkdGJEYxLUkiQkdGJDYkRjRJJnRoZXRhR0Y1RjEhIiIqJkkidUdGJEYxLUklZGlmZkdGNTYkLUkiQUdGJEY5RjRGMUYxKiZGNkYxLUY/NiRGQUY6RjFGMTckNyMiIiMqJiwqKiZGPUYxRjdGMUYxKiZGNkYxRkFGMUYxKihGPUYxLUY/NiRGN0Y0RjFGNEYxRjEqKEY2RjEtRj82JEY3RjpGMUY0RjFGMUYxRjRGOw==

If we write the covariant derivative as the array

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

LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKk9MQF8i

and write the directional covariant derivative as the column vector

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

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqO01AXyI=

we can more easily see that the sum in 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 is well modeled by the product of the matrix with the direction vector.

<Text-field style="Heading 1" layout="Heading 1">Parallelism and Geodesics</Text-field>

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The contravariant vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= defined along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUEYrLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEicEYnL0YzRjEvRjZRJ2l0YWxpY0YnRj5GPkY+, is said to be parallel along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= if

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

the absolute (or intrinsic) derivative along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= vanishes.

Definition 1 Parallelism along a curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

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The ParallelTransportEquations command in the Tensor package generates the equations implied by Definition 1.

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<Text-field style="Heading 2" layout="Heading 2">Example 4</Text-field>

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In polar coordinates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUScmIzk1MjtGJy9GNUZCRj5GPkY+Rj4=, let the curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be given by

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

NyQqJClJInBHNiIiIiMiIiIsJiomRidGKEYlRihGKCIiJEYo

The contravariant vector

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiLUkiQUdGJDYjSSJwR0YkNyQ3IyIiIy1JIkJHRiRGMQ==

is parallel along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= if the equations implied by

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiLCYqKCIiI0YuLUkiQkdGJDYjSSJwR0YkRi4pRjVGMUYuISIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIkFHRiRGNEY1Ri43JDcjRjEqJiwoKihGMUYuRjJGLkY1Ri5GLiomRjFGLkY8Ri5GLiomLUY5NiRGMkY1Ri5GNkYuRi5GLkY2Rjc=

hold. To access the components of this vector and write the individual equations, use

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

LywmKigiIiMiIiItSSJCRzYiNiNJInBHRilGJilGK0YlRiYhIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiQUdGKUYqRitGJiIiIQ==

LywoKigiIiMiIiItSSJCRzYiNiNJInBHRilGJkYrISIiRiYqKEYlRiYtSSJBR0YpRipGJilGK0YlRixGJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRGJ0YrRiYiIiE=

where the extra syntax isolates the two equations in a unique order. Maple provides a solution for these equations and the initial conditions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUYsNiVRIkJGJ0YvRjJGNUZARjpGPg==.

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNictRiw2JVEjQXBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYsNiVRKGNvbWJpbmVGJ0Y2RjktSShtZmVuY2VkR0YkNiQtRiM2JUYrLUYjNiUtRiw2JVEkcmhzRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZhby1GWjYkLUYjNiUtRiw2JVEnc2VsZWN0RidGNkY5LUZaNiQtRiM2KC1GLDYlUSRoYXNGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0Zgby9GVVEsMC4zMzMzMzMzZW1GJy1GLDYlUSRTb2xGJ0Y2RjlGYXAtRiM2JS1GLDYlUSJBRidGNkY5Rl1vLUZaNiQtRiM2Iy1GLDYlUSJwRidGNkY5RkBGK0ZALUZaNiYtRiM2I0YrRkAvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGQEYrRkAtRj02LVEiO0YnRkBGQkZkcEZHRklGS0ZNRk9GYG9GVEYrRis= 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

LCYqJiIiJCIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiI0YlSSJwR0YrRiVGJUYvISIiRiVGJSomRiRGJS1JJHNpbkdGKEYsRiVGJQ==

KiYsJiomIiIkIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJiomIiIjRiZJInBHRixGJkYmRjAhIiJGJkYyKiZGJUYmLUkkY29zR0YpRi1GJkYmRiYpRjFGMEYy

where again, we provide extra syntax for extracting and processing items uniquely. We are looking for some evidence that along the curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= the vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiZUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1JI21uR0YkNiRRIjFGJy9GOVEnbm9ybWFsRicvRjZGNC9GOVEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIitGJ0ZDLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGaW4tRiw2JUYuLUYjNiUtRkA2JFEiMkYnRkNGRUZGRkhGQw== remains parallel. Since the polar plane is coincident with the Cartesian plane, we will look for this evidence in Cartesian coordinates.

We begin by writing the curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in radius (position) vector form:

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

Then, we form the vector 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 and evaluate it along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

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

Figure 1 is a graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= along with vectors that have been transported parallel to the initial vector. Even without the graph it's clear from the algebraic expression for the field along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= that it's constant in the Euclidean sense, and hence, parallel.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Figure 1 Parallel field along the curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

Definition 2 Parallel vector field

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Example 5</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The covariant derivative of the vector field

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

LUkkX0RHRzYiNiM3JDclUSd2ZWN0b3JGJEkiUEdGJDciNyQ3JDcjIiIiLUkiQUdGJDYkSSJyRyUqcHJvdGVjdGVkR0kmdGhldGFHRjM3JDcjIiIjLUkiQkdGJEYx

is, in array form,

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LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKmtMRV8i

The vanishing of this covariant derivative requires solving a set of four partial differential equations, the general solution of which is

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

PCQvLUkiQUc2IjYkSSJyRyUqcHJvdGVjdGVkR0kmdGhldGFHRiksJiomSSRfQzFHRiYiIiItSSRjb3NHNiRGKUkoX3N5c2xpYkdGJjYjRipGLiEiIiomSSRfQzJHRiZGLi1JJHNpbkdGMUYzRi5GLi8tSSJCR0YmRicqJiwmKiZGLUYuRjdGLkYuKiZGNkYuRi9GLkYuRi5GKEY0

For any initial direction of the vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= as a vector in polar coordinates, the equivalent vector in Cartesian coordinates is

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

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqV01FXyI=

Hence, we have a constant vector field, parallel in the Euclidean plane.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be arc length along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the curve defined by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUEYrLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEic0YnL0YzRjEvRjZRJ2l0YWxpY0YnRj5GPkY+. If the unit tangent vector along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is parallel along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a geodesic. The condition that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= must satisfy is

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

Definition 3 Geodesics

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Example 6</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The equations for geodesic curves in polar coordinates are

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

LywmKiYpLUklZGlmZkclKnByb3RlY3RlZEc2JC1JJnRoZXRhR0YoNiNJInNHNiJGLSIiIyIiIi1JInJHRihGLEYwISIiLUYnNiQtRic2JEYxRi1GLUYwIiIh

LywmKioiIiMiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkickdGKTYjSSJzRzYiRi5GJi1GKDYkLUkmdGhldGFHRilGLUYuRiZGKyEiIkYmLUYoNiRGMEYuRiYiIiE=

If we use the overdot to express differentiation with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, we can write these equations as

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

These equations are consistent with Definition 3, as we see by recalling the Christoffel symbols in Table 5. From Definition 3 we have

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

In the first equation, 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 and the remaining Christoffel symbols are zero; in the second, 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 and the other Christoffel symbols are zero.

The general solution of the geodesic equations in polar coordinates is

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnZHNvbHZlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEjREVGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic=

NiQ3JDwjLy1JJnRoZXRhRyUqcHJvdGVjdGVkRzYjSSJzRzYiSSRfQzNHRis8Iy8tSSJyR0YoRiksJiomSSRfQzFHRisiIiJGKkY0RjRJJF9DMkdGK0Y0NyQ8Iy9GJiwmLUknYXJjdGFuRzYkRihJKF9zeXNsaWJHRis2IywkKigjRjQiIilGNCwmKigiIiNGNClGNUZFRjRGKkY0RjQqKEZFRjRGLEY0RkZGNEY0RjRGNSEiIkY0RjRJJF9DNEdGK0Y0PCMvRi8qJkYzRjQpLUklZGlmZkdGKDYkRiZGKiNGNEZFRkg=

The first solution is a radial line segment; the second, an arbitrary line segment - both in accord with our expectations for geodesics in the plane.

<Text-field style="Heading 1" layout="Heading 1">Curvature</Text-field>

JSFH

The commutator formula for covariant differentiation is

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

where

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

and 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. The mixed rank-four tensor 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 is generally called the Riemann-Christoffel curvature tensor of the second kind. Other names include curvature tensor, Riemann tensor, Riemann-Christoffel tensor, and mixed Riemann-Christoffel tensor. Moreover, variations in the formula itself appear in the literature, especially when the repeated covariant differentiation 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 is denoted by 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 or 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. There is an inherent reversal of the lexical order with respect to the operator order. When the reversed lettering is translated to 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, a minus sign is introduced into the definition. The reader is cautioned to be most careful when comparing Maple to the literature.

JSFH

The Riemann-Christoffel curvature tensor of the first kind is defined as 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 and is sometimes called the covariant curvature tensor, and even the Riemann tensor. Again, great care must be taken when reading and comparing different texts.

The Riemann curvature tensor (see how easily these phrases creep into one's writing?) measures the curvature of a space. A space in which this tensor is zero is called flat, and the Cartesian plane is flat no matter what coordinate system is imposed, as we see from

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEwQ3VydmF0dXJlVGVuc29yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJnRidGNkY5RkBGK0Yr

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiUEdGJDckNyZRKGNvbl9iYXNGJFEoY292X2Jhc0YkRi1GLTciNyM3JDcmIiIiRjJGMkYyIiIh

the curvature tensor for the plane under polar coordinates.

A more interesting surface is that of the unit sphere centered at the origin. The metric tensor for this surface is obtained by the following calculations.

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyRRKGNvdl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJJHBoaUdGOkY0RjE=

Consequently, the Riemann-Christoffel curvature tensor of the second kind is

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEqUkMyc3BoZXJlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEwQ3VydmF0dXJlVGVuc29yRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEjZ1NGJ0YvRjJGOQ==

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyZRKGNvbl9iYXNGJFEoY292X2Jhc0YkRi1GLTciNyY3JDcmIiIiIiIjRjJGMyokKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSRwaGlHRjlGM0YyNyQ3JkYyRjNGM0YyLCRGNCEiIjckNyZGM0YyRjJGM0ZANyQ3JkYzRjJGM0YyRjI=

or better yet,

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

NyY3JDcmIiIiIiIjRiVGJiokKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRwaGlHRixGJkYlNyQ3JkYlRiZGJkYlLCRGJyEiIjckNyZGJkYlRiVGJkY0NyQ3JkYmRiVGJkYlRiU=

From this, we construct the explicit representation in Table 10.

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

Table 10 Nonzero components of 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 for the sphere

JSFH

The Riemann-Christoffel curvature tensor of the first kind is

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyZRKGNvdl9iYXNGJEYsRixGLDciNyY3JDcmIiIiIiIjRjFGMiokKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSRwaGlHRjhGMkYxNyQ3JkYxRjJGMkYxLCRGMyEiIjckNyZGMkYxRjFGMkY+NyQ3JkYyRjFGMkYxRjM=

or better yet,

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

NyY3JDcmIiIiIiIjRiVGJiokKS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRwaGlHRixGJkYlNyQ3JkYlRiZGJkYlLCRGJyEiIjckNyZGJkYlRiVGJkYzNyQ3JkYmRiVGJkYlRic=

From this, we construct Table 11.

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

Table 11 Nonzero components of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJWlqaGtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn for the sphere

JSFH

Notice that for a space of dimension LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk= there is just one distinct component needed to describe the complete tensor, and this component is generally taken as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFElMTIxMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+. In general, there are 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 distinct components of the Riemann curvature tensor of the first kind, so for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5Rjk= there 6, but for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjRGJ0Y5Rjk= there are 20.

In the DifferentialGeometry package, the Ricci tensor is defined as 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. Obtained by this definition, the tensor is

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyRRKGNvdl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJJHBoaUdGOkY0RjE=

Obtained with the RicciTensor command, this tensor is

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyRRKGNvdl9iYXNGJEYsNyI3JDckNyQiIiJGMUYxNyQ3JCIiI0Y0KiQpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJJHBoaUdGOkY0RjE=

We see that the tensor is the same under either computation.

The Ricci scalar, given by

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEsUmljY2lTY2FsYXJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRI2dTRidGNkY5RkBGK0Yr

IiIj

is defined as 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, a definition we can test via

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

IiIj

For a two-dimensional manifold the Gaussian (or total) curvature is given by 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the determinant of the array representing the metric tensor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2lqRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==. Since for the sphere 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= can be computed in the Tensor package via

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

KiYpLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJHBoaUdGJyIiIyIiIi1JJF9ER0dGKTYjNyQ3JVEndGVuc29yRilJIlNHRik3JDciNyM3JFEkYmFzRilGLDcjNyRGNiokRiNGLSEiIg==

Finally, we note that 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 is the Einstein tensor. For the sphere, this tensor is

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEvRWluc3RlaW5UZW5zb3JGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRI2dTRidGNkY5RkBGK0Yr

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyRRKGNvbl9iYXNGJEYsNyI3IzckNyQiIiJGMSIiIQ==

so that it vanishes, as we can verify by inspection after obtaining LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImpGJ0YyRjVGMkY1LUYjNiUtRi82JVEiaUYnRjJGNUYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkQvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnL0Y2USdub3JtYWxGJw== via

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

LUkkX0RHRzYiNiM3JDclUSd0ZW5zb3JGJEkiU0dGJDckNyRRKGNvbl9iYXNGJFEoY292X2Jhc0YkNyI3JDckNyQiIiJGMkYyNyQ3JCIiI0Y1RjI=

Since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk=, and the array form of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImpGJ0YyRjVGMkY1LUYjNiUtRi82JVEiaUYnRjJGNUYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkQvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnL0Y2USdub3JtYWxGJw== is the identity matrix, 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 is the zero matrix.JSFH

Legal Notice: \302\251 Maplesoft, a division of Waterloo Maple Inc. 2009. Maplesoft and Maple are trademarks of Waterloo Maple Inc. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact Maplesoft for permission if you wish to use this application in for-profit activities.

JSFH

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