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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 Controls the display of vectors and vector fields, either as column vectors or as the default sum-of-components-times-basis-vectors. Sets the ambient coordinate system and the names of coordinate variables.

 Basic Vector Objects and Constructors 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. 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. A Cartesian vector rooted at the origin, the PositionVector provides a representation of a point, curve, or surface. 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. 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". 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. . The period (or in Typeset math, $·$ from the Common Symbols palette), is the infix dot-product operator between two vectors; alternatively, the DotProduct command. &x The infix cross-product operators are &x and, in Typeset math, $×$ from the Common Symbols palette; alternatively, the CrossProduct command. Conversions between the Cartesian free vector, RootedVector and PositionVector. Evaluates a VectorField at a point, thereby creating a RootedVector with the specified point as the root point. Change coordinates in a free vector or VectorField. Calculates the $p$-norm of a free vector or a VectorField. It defaults to the Euclidean norm. 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 Returns, in a human-readable format, the type and all other relevant data of a Student[VectorCalculus] object 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. Returns information about an object described by a PositionVector. Can be used to give a representation of the object in different coordinate systems. Returns as a free vector, the root point of the vector space to which a RootedVector belongs. 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. Determines whether or not an object is a PositionVector. Determines whether or not an object is a RootedVector. Determines whether or not an object is a VectorField.

 Graphing Graphs the arrows of a VectorField, and one or more flow lines emanating from specified points in the field. The "output = animation" option returns an animation showing the direction in which the flow unfolds. Graphs the curve or surface defined by a PositionVector. Various options permit a variety of vectors and VectorFields to be included in the graph. Graphs free, RootedVectors, and VectorFields. Graphs a curve defined by a free vector. Interactive implementation of the SpaceCurve command, in a setting that also graphs the objects of the Frenet-Serret formalism, including the osculating circle. Both creates the VectorField, and draws its arrows. Interactive implementation of the visual aspects of the VectorField command.





 Frenet-Serret Formalism for Curve Analysis 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. For a curve defined by a free vector or PositionVector, returns a vector tangent to the curve. For a normalization, include the option "normalized". For a curve defined by a free vector or PositionVector, returns a vector along the PrincipalNormal. For a normalization, include the option "normalized". 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. 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 Maple's top-level differential operator modified to map onto the components of vectors. Maple's top-level diff operator modified to map onto the components of vectors. Maple's top-level series operator modified to map onto the components of vectors. $\nabla$ Typeset (2-D math mode) representation of the vector differential operator used in the definitions of gradient, divergence, curl, and Laplacian. In linear (1-D math) input, use "Del" or "Nabla". $\nabla$ In linear (1-D math) input, use "Gradient" to obtain the result of applying the Nabla to a scalar. In Typeset (2-D math) input, apply the symbol $\nabla$ to the scalar. (See the Common Symbols palette for this notation.) $\nabla ·$ In linear (1-D math) input, apply "Divergence" to a VectorField to obtain the divergence of that field. In Typeset (2-D math) input, apply the symbols $\nabla ·$ to the field. (See the Common Symbols palette for these symbols.) $\nabla ×$ In linear (1-D math) input, apply "Curl" to a VectorField to obtain the curl of that field. In Typeset (2-D math) input, apply the symbols $\nabla ×$ to the field. (See the Common Symbols palette for these symbols.) ${\nabla }^{2}$ In linear (1-D math) input, apply "Laplacian" to a scalar to obtain its Laplacian. In Typeset (2-D 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.) 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. Returns , the matrix of second partial-derivatives 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. For a list or vector (free, PositionVector, VectorField), returns either the Jacobian matrix, or a sequence of the Jacobian matrix and its determinant.



 Integration The top-level int command is modified to return the integral of a scalar over pre-defined domains. The inert integral is returned by including the option "output = integral". This command also maps automatically onto the components of a vector. Returns the inert arc-length integral, its value, or a graph of the function, the integrand of the arc-length integral, and the arc-length function. Returns the line integral of the tangential component of a VectorField. Returns the line integral of a scalar. Integrates, along a curve or surface, the normal component of a VectorField. Integrates a scalar on a surface. Returns the scalar whose Gradient is a given VectorField. Returns (unique up to the addition of a Gradient), a VectorField whose Curl is a given VectorField.





 Miscellaneous Returns the parametric equations for a line tangent to a given curve; the parametric equations are the components of a vector. Returns the parametric equations for a plane tangent to a given surface; the parametric equations are the components of a vector. The top-level limit command is modified to map onto the components of a vector.