Building on the Wolfram Language's powerful capabilities in calculus and algebra, the Wolfram Language supports a variety of vector analysis operations. Vectors in any dimension are supported in common coordinate systems. By exploiting the Wolfram Language's efficient representation of arrays, operations can be performed on scalars, vectors, and higher-rank tensors in a uniform manner.

Vector Calculus

Grad () — gradient

Div () — divergence

Curl () — curl in any dimension

Laplacian () — Laplacian

Coordinate Systems

CoordinateChartData — properties of coordinate systems

CoordinateTransformData — relationships between coordinate systems

TransformedField — transform a scalar, vector, or tensor field between coordinate systems

CoordinateTransform — re-express a point in a new coordinate system

FromPolarCoordinates — convert from polar or hyperspherical to Cartesian coordinates

ToPolarCoordinates — convert from Cartesian to polar or hyperspherical coordinates

FromSphericalCoordinates — convert from spherical to Cartesian coordinates

ToSphericalCoordinates — convert from Cartesian to spherical coordinates

Curves

ArcLength — length

ArcCurvature — curvature

FrenetSerretSystem — generalized curvatures and associated basis

LineIntegrate — line integral of a vector field

NLineIntegrate — numerical line integral of a vector field

Higher-Dimensional Parametric Regions

Area — area

Volume — volume

RegionMeasure — arbitrary-dimensional volume

SurfaceIntegrate — surface integral of a vector field

NSurfaceIntegrate — numerical surface integral of a vector field

Visualization

StreamPlot  ▪  VectorPlot  ▪  ListVectorPlot  ▪  VectorPlot3D  ▪  ...