Student[VectorCalculus] Glossary of Commands
The purpose of this help page is to provide a quick reference to the commands in the Student[VectorCalculus] subpackage.
The following table lists all of the commands of the Student[VectorCalculus] subpackage. The commands are categorized according to which of the main components they represent.
For a comprehensive description of this subpackage, see Student[VectorCalculus]
Settings

BasisFormat

Controls the display of vectors and vector fields, either as column vectors or as the default sumofcomponentstimesbasisvectors.

SetCoordinates

Sets the ambient coordinate system and the names of coordinate variables.



Basic Vector Objects and Constructors

<,>
<>
Vector

Constructors for free vectors, the first a column vector, the second, a row vector. Vectors formed with the bracket notation inherit the ambient coordinate system. The Vector command supports declaration of a coordinate system other than the ambient.

The free vector

Vectors formed by any of the three constructors above are called "free vectors." In Cartesian coordinates, the free vector is the arrow from the origin to a point, and is identified with that point. In nonCartesian coordinates, the "free vector" is merely a representation of a point.

PositionVector

A Cartesian vector rooted at the origin, the PositionVector provides a representation of a point, curve, or surface.

RootedVector

A special form of the free vector that is rooted at a given root point. If interpreted as an arrow, the tail of the arrow coincides with the root point.

VectorField

Provides a representation of a mapping from ${\mathrm{\ℝ}}^{n}$ to ${\mathrm{\ℝ}}^{n}$, where $n\=2$ or 3. Thus, the VectorField associates with each point in ${\mathrm{\ℝ}}^{n}$ an arrow whose tail coincides with that point. A graph of representative arrows can be obtained with the option "output = plot".

VectorSpace

Creates a module representing the vector space rooted at a specified point. This space consists of all vectors rooted at that point.



Vector Algebra

+, 

Addition and subtraction of vectors.

*

Multiplication of a vector by a scalar.

.
DotProduct

The period (or in Typeset math, $\xb7$ from the Common Symbols palette), is the infix dotproduct operator between two vectors; alternatively, the DotProduct command.

&x
CrossProduct

The infix crossproduct operators are &x and, in Typeset math, $\times$ from the Common Symbols palette; alternatively, the CrossProduct command.

ConvertVector

Conversions between the Cartesian free vector, RootedVector and PositionVector.

evalVF

Evaluates a VectorField at a point, thereby creating a RootedVector with the specified point as the root point.

MapToBasis

Change coordinates in a free vector or VectorField.

Norm

Calculates the $p$norm of a free vector or a VectorField. It defaults to the Euclidean norm.

Normalize

Divides a free vector or VectorField by its $p$norm, returning a vector or VectorField of length 1. Defaults to normalization by the Euclidean norm.



${}$
${}$
Queries

About

Returns, in a humanreadable format, the type and all other relevant data of a Student[VectorCalculus] object

GetCoordinates

Applied to a free vector, RootedVector, or VectorField, returns the coordinate system attributed to that object. Applied without an argument, returns the ambient coordinate system.

GetPVDescription

Returns information about an object described by a PositionVector. Can be used to give a representation of the object in different coordinate systems.

GetRootPoint

Returns as a free vector, the root point of the vector space to which a RootedVector belongs.

GetSpace

Applied to a RootedVector, returns a module that encodes the VectorSpace to which it belongs. The exports of the module are GetCoordinates, GetRootPoint, and Vector.

IsPositionVector

Determines whether or not an object is a PositionVector.

IsRootedVector

Determines whether or not an object is a RootedVector.

IsVectorField

Determines whether or not an object is a VectorField.



${}$
${}$
FrenetSerret Formalism for Curve Analysis

TNBFrame

For a curve defined by a free vector or PositionVector, returns a sequence containing the normalized TangentVector, PrincipalNormal vector, and Binormal vector. Can also graph and animate the curve and one or more of the three Frenet vectors along the curve.

TangentVector

For a curve defined by a free vector or PositionVector, returns a vector tangent to the curve. For a normalization,
include the option "normalized".

PrincipalNormal

For a curve defined by a free vector or PositionVector, returns a vector along the PrincipalNormal. For a normalization,
include the option "normalized".

Binormal

For a curve defined by a free vector or PositionVector, returns a vector along the Binormal. For a normalization,
include the option "normalized".

Curvature

For a curve defined by a free vector or PositionVector, returns the curvature.

RadiusOfCurvature

For a curve defined by a free vector or PositionVector, returns the reciprocal of the curvature.

Torsion

For a curve defined by a free vector or PositionVector, returns the torsion.



${}$
${}$
Differentiation

D

Maple's toplevel differential operator modified to map onto the components of vectors.

diff

Maple's toplevel diff operator modified to map onto the components of vectors.

series

Maple's toplevel series operator modified to map onto the components of vectors.

$\nabla$
Del
Nabla

Typeset (2D math mode) representation of the vector differential operator used in the definitions of gradient, divergence, curl, and Laplacian.
In linear (1D math) input, use "Del" or "Nabla".

$\nabla$
Gradient

In linear (1D math) input, use "Gradient" to obtain the result of applying the Nabla to a scalar. In Typeset (2D math) input, apply the symbol $\nabla$ to the scalar. (See the Common Symbols palette for this notation.)

$\nabla \xb7$
Divergence

In linear (1D math) input, apply "Divergence" to a VectorField to obtain the divergence of that field. In Typeset (2D math) input, apply the symbols $\nabla \xb7$ to the field. (See the Common Symbols palette for these symbols.)

$\nabla \times$
Curl

In linear (1D math) input, apply "Curl" to a VectorField to obtain the curl of that field. In Typeset (2D math) input, apply the symbols $\nabla \times$ to the field. (See the Common Symbols palette for these symbols.)

${\nabla}^{2}$
Laplacian

In linear (1D math) input, apply "Laplacian" to a scalar to obtain its Laplacian. In Typeset (2D math) input, apply the symbol ${\nabla}^{2}$ to the scalar. (Place the exponent "2" on the symbol ∇, which can be found in the Common Symbols palette.)

DirectionalDiff

Applied to a scalar, returns the directional derivative in a given direction. Applied to a VectorField, returns the product of the Jacobian matrix and the direction vector, a representation of the covariant derivative contracted with the direction vector.

Hessian

Returns $\left[\frac{{\partial}^{2}}{\partial {x}_{j}\partial {x}_{i}}f\right]$, the matrix of second partialderivatives of the scalar $f\left({x}_{1}\,\dots \,{x}_{n}\right)$. For a sufficiently differentiable $f$, this matrix is necessarily symmetric. Otherwise the displayed element is the $\left(i\,j\right)$ entry in the matrix.

Jacobian

For a list or vector (free, PositionVector, VectorField), returns either the Jacobian matrix, or a sequence of the Jacobian matrix and its determinant.



${}$
Integration

int

The toplevel int command is modified to return the integral of a scalar over predefined domains. The inert integral is returned by including the option "output = integral". This command also maps automatically onto the components of a vector.

ArcLength

Returns the inert arclength integral, its value, or a graph of the function, the integrand of the arclength integral, and the arclength function.

LineInt

Returns the line integral of the tangential component of a VectorField.

PathInt

Returns the line integral of a scalar.

Flux

Integrates, along a curve or surface, the normal component of a VectorField.

SurfaceInt

Integrates a scalar on a surface.

ScalarPotential

Returns the scalar whose Gradient is a given VectorField.

VectorPotential

Returns (unique up to the addition of a Gradient), a VectorField whose Curl is a given VectorField.



${}$
${}$
Miscellaneous

TangentLine

Returns the parametric equations for a line tangent to a given curve; the parametric equations are the components of a vector.

TangentPlane

Returns the parametric equations for a plane tangent to a given surface; the parametric equations are the components of a vector.

limit

The toplevel limit command is modified to map onto the components of a vector.



${}$
${}$
${}$
${}$
${}$
${}$
${}$
${}$
${}$
