Overview of the Physics:Vectors Subpackage

Description


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The Physics:Vectors subpackage introduces an algebraic (not matricial) representation for abstract vectors, that is, vectors or vector functions not projected onto any particular vector basis, as well as for cartesian, cylindrical and spherical curvilinear unit vectors, so that it is possible to do algebraic and differential calculus with both nonprojected and projected vector functions. For examples of the use of the subpackage in applications see Physics, examples (this page opens only in the Standard Graphical User Interface).

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The vectors represented in the Vectors subpackage are the standard mathematical objects used in Physics that have magnitude and direction and are defined up to parallel translation, sometimes referred to as free vectors. These vectors can also represent 3Dvectorial noncommutative quantum operators  see for instance the Quantum Mechanics section, of Physics, examples.

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Regarding projected vectors, the Vectors subpackage is designed to work only with cartesian, cylindrical and spherical orthonormal basis and the related systems of coordinates (see Identify), according to

($\mathrm{\_i},\mathrm{\_j},\mathrm{\_k}$)

=

cartesian unit vectors,

($\mathrm{\_\ρ},\mathrm{\_\φ},\mathrm{\_k}$)

=

cylindrical unit vectors,

($\mathrm{\_r},\mathrm{\_\θ},\mathrm{\_\φ}$)

=

spherical unit vectors




($x,y,z$)

=

cartesian coordinates,

($\mathrm{\rho},\mathrm{\phi},z$)

=

cylindrical coordinates,

($r,\mathrm{\theta},\mathrm{\phi}$)

=

spherical coordinates




where $\mathrm{\theta}$ is the angle down from the zaxis, ranging from $0$ to $\mathrm{\pi}$, and $\mathrm{\phi}$ is the angle around the zaxis ranging from $0$ to $2\mathrm{\pi}$.

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Regarding nonprojected vectors and vector functions, the key idea is to identify them by a postfix in the name, as a computer mimicry of "the arrow on top" you use to represent them when working with paper and pencil. This postfix identifier is by default the underscore _ but can be set to be any valid sequence of characters (see Physics[Setup]).

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Mathematical vector notation: When Physics:Vectors is loaded in the Standard Graphical User Interface, and the Typesetting level is set to Extended (the default), nonprojected vectors and unit vectors are respectively displayed with an arrow and a hat on top, and the differential operators (Nabla, Laplacian, etc.) with an upside down triangle as in textbooks. You can also set this notation by entering Physics[Setup](mathematicalnotation = true). You can also set this notation from the Options Dialog: go to Tools > Options, select the Display tab, and set the Typesetting level to Extended.


NOTE: these variables x, y, z, $\mathrm{\rho}$, $\mathrm{\phi}$, r and $\mathrm{\theta}$, as well as _i, _j, _k, $\mathrm{\_\ρ}$, $\mathrm{\_\φ}$, _r and $\mathrm{\_\θ}$, respectively used to represent the coordinates and the unit vectors, are automatically protected when the Physics:Vectors subpackage is loaded.



List of Physics:Vectors Subpackage Commands


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The following is a list of available commands.

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Inert forms of these commands, representing the operations, including their mathematical properties under differentiation, expansion etc., but holding the computations, consist of the same command's names prefixed by the % character. The inert computations constructed with these commands can be activated when desired using the value command.


Brief description of each command


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+, , . and &x performs the addition, subtraction, dot product and cross product of vector functions, respectively.

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ChangeBasis changes the projection basis of a given vector function.

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ChangeCoordinates changes the coordinates in an expression to one of cartesian, cylindrical and spherical coordinates.

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Component evaluates the component of a vector function.

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diff is a differentiation command which takes into account the relation between geometrical cartesian, cylindrical and spherical coordinates (and curvilinear unit vectors) when evaluating derivatives. It uses the same syntax (calling sequence) and display as the standard diff command, (diff and diff derivatives evaluated over one function at the same time are displayed separately).

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DirectionalDiff evaluates the directional derivative of a (vectorial) expression.

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Gradient, Divergence, Curl, and Laplacian, as well as the corresponding inert forms (starting with %) respectively compute (or represent) the gradient, divergence, curl, and Laplacian of a given (vector) function. As a handy mnemonics rule, these differential commands as well as the main one, Nabla (for nabla), are entered using just the first three letters.

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Identify identifies the vector character of an expression in the context of the Physics:Vectors package and presents the package's conventions.

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int is an extension of the int command to additionally compute path, surface and volume integrals from the mathematical representation of the region, vectorial or parametric.

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Nabla is a command representation for the nabla differential operator. Thus, Nabla alone can also be used to calculate the gradient, divergence, curl or Laplacian of a function.

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Norm evaluates the norm of a vector functions (note that Maple has also a command called norm not related to Vectors).

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ParametrizeCurve parametrizes, in terms of a single parameter, curves given in vectorial or algebraic form, or passed as a set of algebraic equations, possibly unsolved.

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ParametrizeSurface parametrizes, in terms of two parameters, surfaces given in vectorial or algebraic form, or passed as a set of algebraic equations, possibly unsolved.

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ParametrizeVolume parametrizes, in terms of three parameters, volumes given in vectorial or algebraic form, or passed as a set of algebraic equations, possibly unsolved.




References



ChebTerrab, E.S. and Nisembaum, M. "Vector Analysis and Symbolic Computation in Physics Education." Workshop: Computers in Education, EDAI  UERJ. Rio de Janeiro, Brazil, 1995.


