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Overview of the VectorCalculus Package

Basic Functionality

Description

 • The VectorCalculus package is a collection of commands that perform multivariate and vector calculus operations.
 Multivariate calculus refers to the calculus of functions from ${R}^{n}$ to $R$. Vector calculus refers to the calculus of functions from ${R}^{n}$ to ${R}^{m}$, where $1.
 • The VectorCalculus package contains a large set of predefined coordinate systems. All computations in the package can be performed in any of these coordinate systems.  By default, the Cartesian coordinate system is used.  The basic objects on which the commands in the VectorCalculus package operate are Vectors, vector fields, and scalar functions.
 • For a complete list of the routines in the VectorCalculus package and advanced information on the capabilities of this package, see the Details of the VectorCalculus package help page.

Interfaces to the VectorCalculus Package

Commands

 • Each command in the VectorCalculus package can be accessed by using either the long form or the short form of the command name in the command calling sequence.  For more information, see the Using Packages help page.
 Long form
 > VectorCalculus:-CrossProduct(, );
 Short form
 > with(VectorCalculus):
 > CrossProduct(, );

 • Some routines in the VectorCalculus package come with a task template to step you through the process of solving a vector calculus problem. For more information, see the Using Tasks help page.

Student:-VectorCalculus Package

 • For students learning the concepts presented in an introductory vector calculus course, see the Student:-VectorCalculus help page.

Essential VectorCalculus Package Commands

 returns information about a VectorCalculus object computes the cross product of Vectors and differential operators compute the curl of a vector field in R^3 computes the directional derivative of a scalar field in the direction given by a vector evaluate a vector field at a point compute the flux of a vector field through a surface in R^3 or a curve in R^2 compute the Laplacian of functions from R^n to R, or of a vector field compute the line integral of a vector field in R^n converts Vectors and vector fields Vector differential operator plots a curve or surface creates a position Vector with specified components compute the radius of curvature of a curve creates a Vector rooted at a point with specified components set the coordinate attribute on a free Vector compute the torsion of a curve in R^3 creates a free Vector with specified components creates a vector field

Examples

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$

Compute the cross product.

 > $\mathrm{CrossProduct}\left(⟨a,b,c⟩,⟨d,e,f⟩\right)$

Compute the radius of curvature.

 > $\mathrm{RadiusOfCurvature}\left(⟨2\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)⟩\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}t::\mathrm{real}$
 $\frac{{\left({-}{3}{}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{+}{4}\right)}^{{3}}{{2}}}}{{2}}$ (1)

Integrate a function over R^2.

 > $\mathrm{int}\left({x}^{2}+{y}^{2},\left[x,y\right]=\mathrm{Circle}\left(⟨0,0⟩,r\right)\right)$
 $\frac{{\mathrm{\pi }}{}{{r}}^{{4}}}{{2}}$ (2)

Change the coordinate system to cylindrical.

 > $\mathrm{SetCoordinates}\left({'\mathrm{cylindrical}'}_{r,\mathrm{θ},z}\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{\theta }}{,}{z}}$ (3)

Define a vector field.

 > $F≔\mathrm{VectorField}\left(⟨{r}^{3},\frac{z}{\mathrm{θ}},\sqrt{r}⟩\right)$

Compute the flux of a vector field through a specified surface.

 > $\mathrm{Flux}\left(F,\mathrm{Sphere}\left(⟨0,0,0⟩,R\right)\right)$
 $\frac{{32}{}{{R}}^{{5}}{}{\mathrm{\pi }}}{{15}}$ (4)

Compute the Laplacian of a vector field.

 > $\mathrm{simplify}\left(\mathrm{Laplacian}\left(F\right)\right)$

Details