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

<Text-field style="Heading 1" layout="Heading 1">Directional Derivative of a Scalar</Text-field>

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

<Text-field style="Heading 3" layout="Heading 3">Top Level</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

If 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 is a unit vector in Cartesian coordinates, then the rate of change of the scalar function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEieEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiZRInlGJ0YvRjJGNUYvRkRGL0ZERi9GRA== taken at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJhRicvJSVzaXplR1EjMTBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEiYkYnRjRGN0Y6RjRGQUY0RkFGNEZB in the direction U is

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LCYqJi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIiNiNJImZHRis2JEkiYUdGK0kiYkdGK0YtSSJ1R0YrRi1GLSomLS0mRic2IyIiI0YuRjBGLUkidkdGK0YtRi0=

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

as computed by Maple. This directional derivative can be written as 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, or as 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, provided we define LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEmJkRlbDtGJy8lJXNpemVHUSMxMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xMTExMTExZW1GJy1JI21pR0YkNiZRImZGJ0YvLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YvRjI= as the vector 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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The DirectionalDiff command in the VectorCalculus packages will compute, at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJhRicvJSVzaXplR1EjMTBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEiYkYnRjRGN0Y6RjRGQUY0RkFGNEZB, the directional derivative of the scalar function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEieEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiZRInlGJ0YvRjJGNUYvRkRGL0ZERi9GRA== in the direction of the vector V, where V does not have to be a unit vector. We illustrate these calculations for the Student VectorCalculus package because this package is a bit more lenient with respect to defining coordinate systems and the names of coordinate variables.

Tools\342\211\273Load Package: Student Vector Calculus

Loading Student:-VectorCalculusJSFH

with(Student:-VectorCalculus):

Define the vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2KFEiVkYnLyUlc2l6ZUdRIzEwRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGOi1JI21vR0YkNjBRIj1GJ0YvRjJGOEY7LyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYmUSJ1RidGLy9GNkY0L0Y5USdpdGFsaWNGJy1GPjYuUSJ+RidGLy9GOVEnbm9ybWFsRidGQUZDRkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU0Zqbi1GLDYoUSJpRidGL0YyRjVGOEY7LUY+Ni5RIitGJ0YvRmduRkFGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGY28tRiw2JlEidkYnRi9GV0ZYRlotRiw2KFEiakYnRi9GMkY1RjhGO0YvRmdu.

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

Apply the DirectionalDiff command. Note that Maple normalizes V.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiooLUklZGlmZkdGJjYkLUkiZkdGKDYkSSJ4R0YmSSJ5R0YmRjQiIiIpLCYqJClJInVHRigiIiNGNkY2KiQpSSJ2R0YoRjxGNkY2I0Y2RjwhIiJGO0Y2RjYqKC1GLzYkRjFGNUY2RjdGQUY/RjZGNkYr

The DirectionalDiff command supports evaluation at a point from within the command itself.

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiooLUklZGlmZkdGJjYkLUkiZkdGKDYkSSJhR0YoSSJiR0YoRjQiIiIpLCYqJClJInVHRigiIiNGNkY2KiQpSSJ2R0YoRjxGNkY2I0Y2RjwhIiJGO0Y2RjYqKC1GLzYkRjFGNUY2RjdGQUY/RjZGNkYr

Obtain the directional derivative from first principles

Apply \342\210\207 to obtain the gradient of f.

Apply the Normalize command to V.

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print(); # input placeholder

Normalize V by dividing it by its length.

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMqJiksJiokKUkidUdGKCIiIyIiIkYzKiQpSSJ2R0YoRjJGM0YzI0YzRjIhIiIsJiomRjFGMy1JJWRpZmZHRiY2JC1JImZHRig2JEkieEdGJkkieUdGJkZBRjNGMyomRjZGMy1GPDYkRj5GQkYzRjNGM0Yr

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<Text-field style="Heading 3" layout="Heading 3">Student MultivariateCalculus</Text-field>

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The Student MultivariateCalculus package has a DirectionalDerivative command that normalizes the direction vector and also has the "evaluation" property.

Tools\342\211\273Load Package: Student Multivariate Calculus

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Loading Student:-MultivariateCalculus

with(Student:-MultivariateCalculus):

Apply the DirectionalDerivative command.

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print(); # input placeholder

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<Text-field style="Heading 3" layout="Heading 3">Physics:-Vectors</Text-field>

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The Physics:-Vectors package also has a DirectionalDiff command that will normalize the direction vector. However, it does not have the "evaluation" property.

Restart Maple.

Load the Physics:-Vectors package.

Allow notational enhancements.

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print(); # input placeholder

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Define the direction vector V. The basis vector i can be entered as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjX2lGJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YvL0Y2USdub3JtYWxGJw== or as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiZRImlGJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW9HRiQ2LlEiXkYnRjIvRjlRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlRGMkY1RjgvRlFGN0YyRkE=.

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJWR0YoLCYqJkkjX2lHRiYiIiJJInVHRihGMUYxKiZJI19qR0YmRjFJInZHRihGMUYxNyNGLg==

Apply the DirectionalDiff command. The vector V is normalized by DirectionalDiff.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEwRGlyZWN0aW9uYWxEaWZmRicvJSVzaXplR1EjMTBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiUtRiM2KC1GLDYmUSJmRidGL0YyRjUtRjk2JS1GIzYnLUYsNiZRInhGJ0YvRjJGNS1JI21vR0YkNi5RIixGJ0YvL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZPLyUqc3ltbWV0cmljR0ZPLyUobGFyZ2VvcEdGTy8lLm1vdmFibGVsaW1pdHNHRk8vJSdhY2NlbnRHRk8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYmUSJ5RidGL0YyRjVGL0ZLRi9GS0ZHLUYsNihRIlZGJ0YvLyUlYm9sZEdGNC9GM0ZPL0Y2USVib2xkRicvJStmb250d2VpZ2h0R0Zmb0YvRktGL0ZLRi8vJStleGVjdXRhYmxlR0ZPRks=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMqJiwmKiYtSSVkaWZmR0YmNiQtSSJmR0YoNiRJInhHRiZJInlHRiZGNSIiIkkidUdGKEY3RjcqJi1GMDYkRjJGNkY3SSJ2R0YoRjdGN0Y3KSwmKiQpRjgiIiNGN0Y3KiQpRjxGQUY3RjcjRjdGQSEiIkYr

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

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In Cartesian coordinates, the defining calculation

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print(); # input placeholder

LCYqJi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIiNiNJImZHRis2JEkiYUdGK0kiYkdGK0YtSSJ1R0YrRi1GLSomLS0mRic2IyIiI0YuRjBGLUkidkdGK0YtRi0=

suggests that when the limit of this derivative exists, it can be more easily calculated as 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. The paradigm extends to other coordinate systems.

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Restart Maple.

Tools\342\211\273Load Package: Student Vector Calculus

Loading Student:-VectorCalculusJSFH

with(Student:-VectorCalculus):

Define the direction vector V as a vector field in polar coordinates.

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

print(); # input placeholder

Apply the DirectionalDiff command to 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 The third argument indicates the system in which to interpret LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZkYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi8vRjZRJ25vcm1hbEYn.

Simplify: 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, and 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

Change 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 to 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, etc.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiooKSwmKiQpSSJ1R0YoIiIjIiIiRjQqJClJInZHRihGM0Y0RjQjRjRGMyEiIi1JJWRpZmZHRiY2JC1JImZHRig2JEkickdGKEkmdGhldGFHRihGQEY0RjJGNEY0KipGQEY5Ri5GOS1GOzYkRj1GQUY0RjdGNEY0Ris=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Maple obtains this result by mapping the whole calculation back to Cartesian coordinates, applying the paradigm 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, then mapping back to polar coordinates. Note that Maple also normalizes the vector V.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Physics:-Vectors</Text-field>

JSFH

Like the Student VectorCalculus package, the Physics:-Vectors package can work in Cartesian LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEoJiM4NDc3O0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW5HRiQ2JVEiMkYnRjJGOEYyL0Y2USV0cnVlRicvRjlRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YyRjg= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEoJiM4NDc3O0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW5HRiQ2JVEiM0YnRjJGOEYyL0Y2USV0cnVlRicvRjlRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YyRjg=, as well as spherical and cylindrical coordinates. For polar coordinates, Physics:-Vectors uses the cylindrical system with basis vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiZRJyYjOTYxO0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW9HRiQ2LlEiXkYnRjJGOC8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlFGMi9GNlEldHJ1ZUYnL0Y5USdpdGFsaWNGJy9GTkZVRjJGOA== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiZRJyYjOTY2O0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW9HRiQ2LlEiXkYnRjJGOC8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlFGMi9GNlEldHJ1ZUYnL0Y5USdpdGFsaWNGJy9GTkZVRjJGOA==. The VectorCalculus package itself works with all of the orthogonal coordinate systems known to Maple. (See LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEiP0YnLyUlc2l6ZUdRIzEwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRkYtSSNtaUdGJDYmUSdjb29yZHNGJ0YvLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YvRjI= for the complete list.)

Restart Maple.

Load the Physics:-Vectors package.

Allow notational enhancements.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEocmVzdGFydEYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRjU=

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

Define the direction vector V in polar coordinates. The basis vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiZRJyYjOTYxO0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW9HRiQ2LlEiXkYnRjJGOC8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlFGMi9GNlEldHJ1ZUYnL0Y5USdpdGFsaWNGJy9GTkZVRjJGOA== can be entered as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEoXyYjOTYxO0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi8vRjZRJ25vcm1hbEYn or as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1GIzYmLUkjbWlHRiQ2JlEnJiM5NjE7RicvJSVzaXplR1EjMTBGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjQvRjhRJXRydWVGJy9GO1EnaXRhbGljRictRiM2Ji1JI21vR0YkNi5RIl5GJ0Y0RjovJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZXRjRGPUY/L0ZURj5GNEY6.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJWR0YoLCYqJkkidUdGKCIiIkklX3Job0dGJkYxRjEqJkkidkdGKEYxSSVfcGhpR0YmRjFGMTcjRi4=

Apply the DirectionalDiff command. The vector V is normalized by DirectionalDiff.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMqJiwmKiYtSSVkaWZmR0YmNiQtSSJmR0YoNiRJJHJob0dGJkkkcGhpR0YmRjUiIiJJInVHRihGN0Y3KigtRjA2JEYyRjZGN0Y1ISIiSSJ2R0YoRjdGN0Y3KSwmKiQpRjgiIiNGN0Y3KiQpRj1GQkY3RjcjRjdGQkY8Ris=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

This result is equivalent to the result produced by the DirectionalDiff command in the VectorCalculus packages.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Directional Derivative of a Vector</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The operator 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 acting on a vector B appears in the vector identities for the gradient of a dot product, and for the curl of a cross product. In particular, the operator appears in the identities

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

and

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

and is generally interpreted as the directional derivative of one vector taken in the direction of the other vector. Indeed, it is a form of the directional covariant derivative of differential geometry. Just as the directional derivative of a scalar leads to the definition of the gradient vector, so too does the directional derivative of a vector lead to the covariant derivative, a rank-two tensor.

Implementing this directional derivative of a vector in the confines of the VectorCalculus packages pushes these packages to their limits. In effect, the calculation can only be done in Cartesian coordinates, and that, by an artifact. Outside its DifferentialGeometry package, Maple cannot form the operator 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, so must first compute the gradients of each component of B, and then dot A with each such gradient.

The Physics:-Vectors package extends the applicability of its DirectionalDiff command, but unfortunately normalizes the direction vector when applied to a scalar (correct) and also when applied to a vector (problematic).

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Restart Maple.

Tools\342\211\273Load Package: Student Vector Calculus

Set display format for vectors.

Loading Student:-VectorCalculus

with(Student:-VectorCalculus):

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEsQmFzaXNGb3JtYXRGJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JS1GIzYmLUYsNiZRJmZhbHNlRidGL0YyRjVGLy8lK2V4ZWN1dGFibGVHRj8vRjZRJ25vcm1hbEYnRi9GQi1JI21vR0YkNi5RIjpGJ0YvRkIvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZYRi9GQEZC

Define vector fields F and V.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LUkobWFjdGlvbkc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYpLUklbXJvd0dGJDYnLUknbXRhYmxlR0YkNjYtSSRtdHJHRiQ2Ji1JJG10ZEdGJDYoLUYvNictRi82Jy1GLDYlLUYvNiktSSZtZnJhY0dGJDYoLUYvNictSSNtb0dGJDYuUSsmUGFydGlhbEQ7RicvJSVzaXplR1EjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlMvJSlzdHJldGNoeUdGUy8lKnN5bW1ldHJpY0dGUy8lKGxhcmdlb3BHRlMvJS5tb3ZhYmxlbGltaXRzR0ZTLyUnYWNjZW50R0ZTLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGXG9GSy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGU0ZOLUYvNictRi82KEZHLUkjbWlHRiQ2JlEieEYnRksvJSdpdGFsaWNHUSV0cnVlRicvRk9RJ2l0YWxpY0YnRktGX29GYm9GTkZLRl9vRmJvRk4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmZwLyUpYmV2ZWxsZWRHRlMtRkg2LVExJkludmlzaWJsZVRpbWVzO0YnRk5GUUZURlZGWEZaRmZuRmhuRmpuRl1vLUYvNiktRmlvNiZRImZGJ0ZLRlxwRl9wLUZINi5RMCZBcHBseUZ1bmN0aW9uO0YnRktGTkZRRlRGVkZYRlpGZm5GaG5Gam5GXW8tRiw2JS1GLzYpRmhvLUZINi5RIixGJ0ZLRk5GUS9GVUZecEZWRlhGWkZmbkZobkZqbi9GXm9RLDAuMzMzMzMzM2VtRictRmlvNiZRInlGJ0ZLRlxwRl9wRktGX29GYm9GTkZLRk5GS0Zfb0Zib0ZORktGX29GYm9GTkZLRk5GW3EtRmlvNiZRInVGJ0ZLRlxwRl9wRktGTi1GSDYuUSIrRidGS0ZORlFGVEZWRlhGWkZmbkZobi9GW29RLDAuMjIyMjIyMmVtRicvRl5vRmpyLUYvNictRiw2JS1GLzYpLUZDNihGRS1GLzYnLUYvNihGR0ZgckZLRl9vRmJvRk5GS0Zfb0Zib0ZORmFwRmRwRmdwRmlwRltxRl5xRktGX29GYm9GTkZLRk5GW3EtRmlvNiZRInZGJ0ZLRlxwRl9wRktGTkZLRk4vJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRl10LyUrZ3JvdXBhbGlnbkdGXXQvJShyb3dzcGFuR0ZjcC8lK2NvbHVtbnNwYW5HRmNwRlt0Rl50RmB0LUY1NiYtRjg2KC1GLzYnLUYvNictRiw2JS1GLzYpRkJGW3EtRi82KS1GaW82JlEiZ0YnRktGXHBGX3BGY3FGZnFGS0Zfb0Zib0ZORktGX29GYm9GTkZLRk5GW3FGY3JGS0ZORmZyLUYvNictRiw2JS1GLzYpRmJzRltxRmJ1RktGX29GYm9GTkZLRk5GW3FGaHNGS0ZORktGTkZbdEZedEZgdEZidEZkdEZbdEZedEZgdC8lJmFsaWduR1ElYXhpc0YnL0ZcdFEpYmFzZWxpbmVGJy9GX3RGZnAvRmF0USd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGXnAvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0Zpdi8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0Zkdy8lJmZyYW1lR0Zkdy8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRlMvJS1lcXVhbGNvbHVtbnNHRlMvJS1kaXNwbGF5c3R5bGVHRlMvJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGYXdGS0Zfb0Zib0ZORktGTi9JK21zZW1hbnRpY3NHRiRRKkNvbFZlY3RvckYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmd4LyUrYWN0aW9udHlwZUdRLnJ0YWJsZWFkZHJlc3NGJy8lKXJ0YWJsZWlkR1EpODU5NDI2MDRGJw==

Compute 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 by mapping DirectionalDiff onto the components of F. This then requires multiplying by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYnLUYjNiUtSSNtaUdGJDYoUSJWRicvJSVzaXplR1EjMTBGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y/RjQvRj5RJ25vcm1hbEYnRjRGQi8lJW9wZW5HUSYmcGFyO0YnLyUmY2xvc2VHRkZGNEZC because DirectionalDiff normalizes the direction vector V.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

JSFH

The syntax for mapping the DirectionalDiff command onto the components of F is far more complex than that required for mapping both the gradient and the dot product. This is for two reasons. First, the DirectionalDiff command normalized the direction vector (here, V); and second, because the Norm command in the VectorCalculus packages produces a function whose value at a point is the norm of the given vector.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

A proposal to modify DirectionalDiff in the VectorCalculus packages has already been made. Under this proposal, application of the command to a vector field would be allowed, and when so applied, the direction vector would not be normalized. Maplesoft has given serious consideration to this modification, and if implemented, would allow the (2-D) Math notation 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 to stand for 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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Physics:-Vectors</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Although the DirectionalDiff command in the Physics:-Vectors package automatically maps onto the components of a vector, the direction vector is normalized.

Unlike the VectorCalculus packages, the Physics:-Vectors package lets the syntax 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 to form a valid operator that can be applied to a vector B. Hence, the notation 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 will compute the directional derivative of B in the direction of A.

Restart Maple.

Load the Physics:-Vectors package.

Allow notational enhancements.

Define the vector F. The basis vector i can be entered as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjX2lGJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YvL0Y2USdub3JtYWxGJw== or as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiZRImlGJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW9HRiQ2LlEiXkYnRjIvRjlRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlRGMkY1RjgvRlFGN0YyRkE=.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJGR0YoLCYqJi1JImZHRig2JEkieEdGJkkieUdGJiIiIkkjX2lHRiZGNUY1KiYtSSJnR0YoRjJGNUkjX2pHRiZGNUY1NyNGLg==

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJWR0YoLCYqJkkidUdGKCIiIkkjX2lHRiZGMUYxKiZJInZHRihGMUkjX2pHRiZGMUYxNyNGLg==

Apply the DirectionalDiff command, organizing the result by components.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiooKSwmKiQpSSJ1R0YoIiIjIiIiRjQqJClJInZHRihGM0Y0RjQjRjRGMyEiIiwmKiZGMkY0LUklZGlmZkdGJjYkLUkiZkdGKDYkSSJ4R0YmSSJ5R0YmRkJGNEY0KiZGN0Y0LUY9NiRGP0ZDRjRGNEY0SSNfaUdGJkY0RjQqKEYuRjksJiomRjJGNC1GPTYkLUkiZ0dGKEZBRkJGNEY0KiZGN0Y0LUY9NiRGTUZDRjRGNEY0SSNfakdGJkY0RjRGKw==

Implement the notation 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, again organizing the result by components.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiomLCYqJkkidUdGKCIiIi1JJWRpZmZHRiY2JC1JImZHRig2JEkieEdGJkkieUdGJkY4RjFGMSomSSJ2R0YoRjEtRjM2JEY1RjlGMUYxRjFJI19pR0YmRjFGMSomLCYqJkYwRjEtRjM2JC1JImdHRihGN0Y4RjFGMSomRjtGMS1GMzYkRkRGOUYxRjFGMUkjX2pHRiZGMUYxRis=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Since the DirectionalDiff command again normalizes the direction vector, the best implementation of the directional derivative of a vector is via the notation 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 in the Physics:-Vectors package, at least in Cartesian coordinates.

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

JSFH

As earlier, polar coordinates are used as an example of nonCartesian coordinates.

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

In the VectorCalculus packages, the mapping artifact that works in Cartesian coordinates does not work in polar coordinates. A wrong answer results. Therefore, to obtain the directional derivative of a vector in polar coordinates, the calculation must be implemented in Cartesian coordinates.

JSFH

Restart Maple.

Tools\342\211\273Load Package: Student Vector Calculus

Set display format for vectors.

Loading Student:-VectorCalculus

with(Student:-VectorCalculus):

Define, in polar coordinates, the vector fields F and V.

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Apply the MapToBasis command to convert F and V to vector fields in Cartesian coordinates.

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Obtain the directional derivative.

Map the result back to polar coordinates.

Simplify: 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, and 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print(); # input placeholder

Display the directional derivative, after having grouped the result by components.

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print(); # input placeholder

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

In this directional derivative, the terms 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 and 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 in the first and second components respectively, imply that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2KFEiVkYnLyUlc2l6ZUdRIzEwRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGOi1JI21vR0YkNjBRKSZtaWRkb3Q7RidGL0YyRjhGOy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRlEtRj42LlEoJm5hYmxhO0YnRi8vRjlRJ25vcm1hbEYnRkFGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNRLDAuMTExMTExMWVtRidGL0ZX cannot be represented in nonCartesian coordinates as an operator acting componentwise on F. This is because ultimately, derivatives have to be taken of the varying basis vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiZUYnLyUlc2l6ZUdRIzEwRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPS1GIzYmLUYvNiZRInJGJ0YyL0Y5RjcvRjxRJ2l0YWxpY0YnRjJGRUZGLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMi9GPFEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiZUYnLyUlc2l6ZUdRIzEwRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPS1GIzYmLUYvNiZRJyYjOTUyO0YnRjJGOC9GPFEnbm9ybWFsRidGMi9GOUY3L0Y8USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjJGRQ==. Indeed, let 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, 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, define a line through the Cartesian point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJhRicvJSVzaXplR1EjMTBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEiYkYnRjRGN0Y6RjRGQUY0RkFGNEZB in the direction 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. The polar vector F evaluated along this line is

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The directional derivative is given by 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, in which the basis vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiZUYnLyUlc2l6ZUdRIzEwRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPS1GIzYmLUYvNiZRInJGJ0YyL0Y5RjcvRjxRJ2l0YWxpY0YnRjJGRUZGLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMi9GPFEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiZUYnLyUlc2l6ZUdRIzEwRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPS1GIzYmLUYvNiZRJyYjOTUyO0YnRjJGOC9GPFEnbm9ybWFsRidGMi9GOUY3L0Y8USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjJGRQ== vary along the line described by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEic0YnRi9GMkY1Ri8vRjZRJ25vcm1hbEYnRi9GQC1JI21vR0YkNi5RIixGJ0YvRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEieUYnRi9GMkY1RjhGL0ZA. Thus, computing 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 requires differentiating the products 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiZ0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEifkYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk8tSSVtc3ViR0YkNiUtRiw2KFEiZUYnRi8vJSVib2xkR0Y0L0YzRkAvRjZRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRmZuLUYjNiYtRiw2JlEnJiM5NTI7RidGL0ZaRjxGL0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YvRjw=, a process that would generate the Christoffel symbols of covariant differentiation. It is this differentiation process that is responsible for the non-differentiated terms in the directional derivative, and therefore the need to carry out the calculation in Cartesian coordinates.

JSFH

<Text-field style="Heading 3" layout="Heading 3">Physics:-Vectors</Text-field>

JSFH

As pointed out earlier, the Physics:-Vectors package uses cylindrical coordinates to implement polar coordinates. The basis vectors are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiZRJyYjOTYxO0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW9HRiQ2LlEiXkYnRjJGOC8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlFGMi9GNlEldHJ1ZUYnL0Y5USdpdGFsaWNGJy9GTkZVRjJGOA== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiZRJyYjOTY2O0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbW9HRiQ2LlEiXkYnRjJGOC8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlFGMi9GNlEldHJ1ZUYnL0Y5USdpdGFsaWNGJy9GTkZVRjJGOA==.

Unfortunately, none of 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, or 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 give the correct directional derivative for vectors F and V in polar coordinates. The calculation must still be carried out in Cartesian coordinates, even in the Physics:-Vectors package.

Restart Maple.

Load the Physics:-Vectors package.

Allow notational enhancements.

Define, in polar coordinates, the vector fields F and V.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

The following three results are not correct. The first two are missing the non-differentiated terms. The third, using syntax that worked for the scalar case, has incorrect and missing terms.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiooKSwmKiQpSSJ1R0YoIiIjIiIiRjQqJClJInZHRihGM0Y0RjQjRjRGMyEiIiwmKiZGMkY0LUklZGlmZkdGJjYkLUkiZkdGKDYkSSRyaG9HRiZJJHBoaUdGJkZCRjRGNCooRjdGNC1GPTYkRj9GQ0Y0RkJGOUY0RjRJJV9yaG9HRiZGNEY0KihGLkY5LCYqJkYyRjQtRj02JC1JImdHRihGQUZCRjRGNCooRjdGNC1GPTYkRk1GQ0Y0RkJGOUY0RjRJJV9waGlHRiZGNEY0Ris=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiomLCYqJkkidUdGKCIiIi1JJWRpZmZHRiY2JC1JImZHRig2JEkkcmhvR0YmSSRwaGlHRiZGOEYxRjEqKEkidkdGKEYxLUYzNiRGNUY5RjFGOCEiIkYxRjFJJV9yaG9HRiZGMUYxKiYsJiomRjBGMS1GMzYkLUkiZ0dGKEY3RjhGMUYxKihGO0YxLUYzNiRGRUY5RjFGOEY+RjFGMUklX3BoaUdGJkYxRjFGKw==

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiooLCgtSSVkaWZmR0YmNiQtSSJmR0YoNiRJJHJob0dGJkkkcGhpR0YmRjUiIiIqJkY1ISIiRjJGN0Y3KiYtRjA2JC1JImdHRihGNEY2RjdGNUY5RjdGN0kidUdGKEY3SSVfcmhvR0YmRjdGNyooRi5GN0kidkdGKEY3SSVfcGhpR0YmRjdGN0Yr

JSFH

The correct result can be obtained by the same process used with polar coordinates in the VectorCalculus package. The following calculations bring the vectors to Cartesian coordinates, compute the directional derivative, and return the results to polar coordinates.

Apply the ChangeBasis command to convert F and V to vectors in Cartesian coordinates.

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

Obtain the directional derivative and simplify: 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

Map the result back to polar coordinates.

Force LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JlEnYXJjdGFuRicvJSVzaXplR1EjMTBGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiktSSZtZnJhY0dGJDYoLUYjNistRiw2JlEnJiM5NjE7RidGL0YyRjUtSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlktRiw2JlEkc2luRidGL0YyRjUtRjk2JS1GIzYoLUYsNiZRJyYjOTY2O0YnRi9GMkY1Ri8vJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjVGL0Y1Ri9GYG9GY29GZm9GNS1GIzYoLUYsNiNRIUYnLUYjNigtRiw2JlEmJnJobztGJ0YvRjJGNUZFLUYsNiZRJGNvc0YnRi9GMkY1LUY5NiUtRiM2JS1GLDYmUSYmcGhpO0YnRi9GMkY1Ri9GNUYvRjVGL0Y1RltwRi8vRjNGZW8vRjZRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZlcS8lKWJldmVsbGVkR0Y0RltwRi9GYG9GY29GZm9GNUYvRjUtRkY2LlEoJnNyYXJyO0YnRi9GNUZJRktGTUZPRlFGU0ZVRldGWkZdb0YvRmBvRmNvRmZvRjU=

Change 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 to 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, etc.

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

Display the directional derivative, after having grouped the result by components.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiomLCgqJkkidUdGKCIiIi1JJWRpZmZHRiY2JC1JImZHRig2JEkkcmhvR0YmSSRwaGlHRiZGOEYxRjEqKEY4ISIiSSJ2R0YoRjEtSSJnR0YoRjdGMUY7KihGOEY7RjxGMS1GMzYkRjVGOUYxRjFGMUklX3Job0dGJkYxRjEqJiwoKihGOEY7RjxGMS1GMzYkRj1GOUYxRjEqJkYwRjEtRjM2JEY9RjhGMUYxKihGOEY7RjxGMUY1RjFGMUYxSSVfcGhpR0YmRjFGMUYr

JSFH

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

There is considerable overhead in setting up a calculation of the directional derivative in the DifferentialGeometry package. Moreover, the notation is ill-adapted for math-mode entry or display of the basic objects of the subject, as demonstrated in greater detail in the earlier Reporter article Tensor Calculus with the Differential Geometry Package, available in the Maple Application Center.

If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEidkYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEia0YnRjJGNUY4RjJGNUY4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YyL0Y5USdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEnJiM5NDU7RicvJSVzaXplR1EjMTBGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtRi82JlEia0YnRjIvRjZRJXRydWVGJy9GOVEnaXRhbGljRidGMkZARkIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGOA== are the contravariant components of the vectors F, and V, respectively, then the directional covariant derivative is given by 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, where 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 is the covariant derivative of F. The following manipulations leading to the invocation of the DirectionalCovariantDerivative command generate the directional covariant derivative of F in the direction of V.

Restart Maple.

Load the package and the necessary subpackages.

Avail of notational compression.

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

Define R2 as a Cartesian frame.

Define P as a polar frame.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JlEoREdzZXR1cEYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRjk2Jy1GIzYnLUYsNiZRInhGJ0YvRjJGNS1JI21vR0YkNi5RIixGJ0YvL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZMLyUqc3ltbWV0cmljR0ZMLyUobGFyZ2VvcEdGTC8lLm1vdmFibGVsaW1pdHNHRkwvJSdhY2NlbnRHRkwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYmUSJ5RidGL0YyRjVGL0ZIRi9GSC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZELUYsNiZRI1IyRidGL0YyRjVGL0ZIRi9GSC1GRTYuUSI6RidGL0ZIRkovRk5GTEZPRlFGU0ZVRlcvRlpRLDAuMjc3Nzc3OGVtRicvRmduRmpvLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRmVuLyUmZGVwdGhHRmFwLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZdcDYmRl9wRmJwRmRwL0ZncFElYXV0b0YnRistRjk2JS1GIzYnLUY5NictRiM2Jy1GLDYmUSJyRidGL0YyRjVGRC1GLDYmUSgmdGhldGE7RidGLy9GM0ZMRkhGL0ZIRi9GSEZcb0Zfb0ZELUYsNiZRIlBGJ0YvRjJGNUYvRkhGL0ZIRmVvRi8vJStleGVjdXRhYmxlR0ZMRkg=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

Define the covariant (tangent) basis vectors 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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlEjQjFGJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIzo9RidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTy1JKG1mZW5jZWRHRiQ2Jy1GIzYnLUYsNiZRJERfeEYnRi9GMkY1LUY5Ni5RIixGJ0YvRjxGPi9GQkY0RkNGRUZHRklGSy9GTlEmMC4wZW1GJy9GUVEsMC4zMzMzMzMzZW1GJy1GLDYmUSREX3lGJ0YvRjJGNUYvRjxGL0Y8LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUYsNiNRIUYnRi8vJStleGVjdXRhYmxlR0ZARjw=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNCMUdGKDckSSREX3hHRihJJERfeUdGKDcjNyQtSSRfREdHRig2IzckNyVRJ3ZlY3RvckYoSSNSMkdGKDciNyM3JDcjIiIiRj4tRjQ2IzckRjc3IzckNyMiIiNGPg==

Define the contravariant (gradient) basis vectors 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.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNCMkdGKDckSSNkeEdGKEkjZHlHRig3IzckLUkkX0RHR0YoNiM3JDclUSVmb3JtRihJI1IyR0YoIiIiNyM3JDcjRjpGOi1GNDYjNyRGNzcjNyQ3IyIiI0Y6

Define the metric tensor

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictRiM2Jy1JJW1zdWJHRiQ2JS1GLDYmUSJnRicvJSVzaXplR1EjMTBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Jy1GLDYmUSNpakYnRjdGOkY9RjcvJStleGVjdXRhYmxlR0Y8LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy9GPlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y3RkVGR0ZKRitGN0ZFRkdGSg== for the Euclidean frame.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JlEjZ0VGJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIzo9RidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTy1GLDYmUSdldmFsREdGJ0YvRjJGNS1JKG1mZW5jZWRHRiQ2JS1GIzYnLUYsNiNRIUYnLUYjNilGWi1GIzYnLUYsNiZRI2R4RidGL0YyRjUtRjk2LlElfiZ0fkYnRi9GPEY+RkFGQ0ZFRkdGSUZLL0ZOUSYwLjBlbUYnL0ZRRmJvRltvRi9GPC1GOTYuUSIrRidGL0Y8Rj5GQUZDRkVGR0ZJRksvRk5RLDAuMjIyMjIyMmVtRicvRlFGaG8tRiM2Jy1GLDYmUSNkeUYnRi9GMkY1Rl5vRlxwRi9GPEZaRi9GPEZaRi9GPEYvRjxGWkYvLyUrZXhlY3V0YWJsZUdGQEY8

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNnRUdGKCwmKiZJI2R4R0YoIiIiRjBGMUYxKiZJI2R5R0YoRjFGM0YxRjE3Iy1JJF9ER0dGKDYjNyQ3JVEndGVuc29yRihJI1IyR0YoNyQ3JFEoY292X2Jhc0YoRj43IjckNyQ3JEYxRjFGMTckNyQiIiNGRUYx

The metric tensor for the polar frame is obtained by transforming the Cartesian one

Equations defining the inverse map from polar to Cartesian.

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

Define the inverse map as a DifferentialGeometry object.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShQaGlfaW52R0YoNyQvSSJ4R0YmKiZJInJHRiYiIiItSSRjb3NHRiU2I0kmdGhldGFHRiZGMy9JInlHRiYqJkYyRjMtSSRzaW5HRiVGNkYzNyMtSSRfREdHRig2IzckNyZRL3RyYW5zZm9ybWF0aW9uRig3JDckSSJQR0YoIiIhNyRJI1IyR0YoRkc3IjcjLUknTWF0cml4R0YlNiMvSSQlaWRHRigiKS8rKmUpNyQ3JEYxRjA3JEY6Rjk=

Obtain the metric tensor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZ0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEjaWpGJ0YyRjVGOEYyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjIvRjlRJ25vcm1hbEYn for polar coordinates.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJnR0YoLCYqJkkjZHJHRigiIiJGMEYxRjEqKClJInJHRiYiIiNGMUknZHRoZXRhR0YoRjFGNkYxRjE3Iy1JJF9ER0dGKDYjNyQ3JVEndGVuc29yRihJIlBHRig3JDckUShjb3ZfYmFzRihGQTciNyQ3JDckRjFGMUYxNyQ3JEY1RjUqJEYzRjE=

Connection for polar coordinates.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJDR0YoLCgqKkkickdGJiEiIkkjZHJHRigiIiJJKERfdGhldGFHRihGM0knZHRoZXRhR0YoRjNGMyoqRjBGM0Y1RjNJJERfckdGKEYzRjVGM0YxKipGMEYxRjVGM0Y0RjNGMkYzRjM3Iy1JJF9ER0dGKDYjNyQ3JVErY29ubmVjdGlvbkYoSSJQR0YoNyQ3JVEoY292X2Jhc0YoUShjb25fYmFzRihGQzciNyU3JDclRjMiIiNGSSokRjBGMTckNyVGSUYzRkksJEYwRjE3JDclRklGSUYzRko=

LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEoJkdhbW1hO0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JJW1yb3dHRiQ2JS1GLDYmUSNpakYnRi8vRjNRJXRydWVGJy9GNlEnaXRhbGljRidGL0Y1LUY5NiUtRiw2JlEia0YnRi9GPkZARi9GNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGSS9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRic=, Christoffel symbols (second kind)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEifkYnLyUlc2l6ZUdRIzEwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkZGL0Yyfor polar coordinates: 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, 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

List the nonzero Christoffel symbols: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

NyU3JDclIiIiIiIjRiYqJEkickclKnByb3RlY3RlZEchIiI3JDclRiZGJUYmLCRGKEYqNyQ3JUYmRiZGJUYn

Define the contravariant vector F.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJGR0YoLCYqJkkiQUdGKCIiIkkkRF9yR0YoRjFGMSomSSJCR0YoRjFJKERfdGhldGFHRihGMUYxNyMtSSRfREdHRig2IzckNyVRJ3ZlY3RvckYoSSJQR0YoNyI3JDckNyNGMS1GMDYkSSJyR0YmSSZ0aGV0YUdGJjckNyMiIiMtRjRGQw==

Define the contravariant vector V.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJWR0YoLCYqJkkidUdGKCIiIkkkRF9yR0YoRjFGMSomSSJ2R0YoRjFJKERfdGhldGFHRihGMUYxNyMtSSRfREdHRig2IzckNyVRJ3ZlY3RvckYoSSJQR0YoNyI3JDckNyNGMUYwNyQ3IyIiI0Y0

Apply the DirectionalCovariantDerivative command to obtain the directional derivative of F along V.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

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

Write the directional covariant derivative as a column vector.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

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

Write the covariant derivative of F as an array.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

print(); # input placeholder

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

At first glance, the result produced by the DirectionalCovariantDerivative does not agree with the earlier result

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

This is because the VectorCalculus and Physics:-Vectors packages use unit basis vectors, so components of vectors are the so-called physical components. The DifferentialGeometry package uses the natural basis vectors, which are not necessarily of unit length. Consequently, to make the results agree, consider the following.

Let R be the position vector for polar coordinates.

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Natural basis vectors: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiZUYnLyUlc2l6ZUdRIzEwRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPS1GIzYmLUYvNiZRInJGJ0YyL0Y5RjcvRjxRJ2l0YWxpY0YnRjJGRUZGLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYuUSIsRidGMi9GPFEnbm9ybWFsRicvJSZmZW5jZUdGOi8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlRi4tRiM2Ji1GLzYmUScmIzk1MjtGJ0YyRjhGT0YyRkVGRkZIRjJGTw==

Unit basis vectors: 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1JI21pR0YkNihRInJGJy8lJXNpemVHUSMxMEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRj0tRiM2Ji1JI21vR0YkNi5RIl5GJ0YyL0Y8USdub3JtYWxGJy8lJmZlbmNlR0Y6LyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlhGMi9GOUY3L0Y8USdpdGFsaWNGJy9GVUY3RjJGRg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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 = 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

Express 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 with unit vectors 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

Express 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 with unit vectors 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

Match physical components with "natural" components.

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

Replace physical components and unit basis vectors with natural components and natural basis vectors.

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

Thus, using either the VectorCalculus or Physics:-Vectors packages to compute 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 in polar coordinates by mapping back to Cartesian coordinates produces the same result as computing the directional covariant derivative in the DifferentialGeometry package. Perhaps one day these Maple packages will become more interconnected.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=