As of Version 9.0, vector analysis functionality is built into the Wolfram Language

Vector Analysis Package

Frequently, physical systems exhibit special symmetries or structures that make a particular coordinate system especially useful. In a mathematically elegant solution to problems related to these systems, often the main step is choosing the correct coordinates. A variety of tools for doing calculus in various threedimensional coordinate systems are provided in this package.

ReferenceReference

SetCoordinates set the default coordinate system

    

CoordinateSystem default coordinate system

Coordinates default variables in the default coordinate system

Coordinate Systems »

Bipolar  ▪  Bispherical  ▪  Cartesian  ▪  ...

Parameters

CoordinateRanges interval over which the coordinate variables range

Parameters default parameters of the coordinate system

ParameterRanges range of values of the parameters of the coordinate system

Coordinate Transformations

CoordinatesToCartesian transform coordinates to Cartesian coordinates

CoordinatesFromCartesian transform coordinates from Cartesian coordinates

Vector Calculus

Div  ▪  Curl  ▪  Grad  ▪  Laplacian  ▪  Biharmonic

ArcLengthFactor derivative of the arc length along parameterized curve

JacobianMatrix derivative matrix of the coordinate transformation

JacobianDeterminant determinant of the Jacobian matrix

ScaleFactors list the scale factors

DotProduct  ▪  CrossProduct  ▪  ScalarTripleProduct