# Vector calculus

**Vector calculus** is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. It consists of a suite of formulas and problem solving techniques very useful for engineering and physics.

We consider vector fields, which associate a vector to every point in space, and scalar fields, which associate a scalar to every point in space. For example, the temperature of a swimming pool is a scalar field: to each point we associate a scalar value of temperature. The water flow in the same pool is a vector field: to each point we associate a velocity vector.

Three operations are important in vector calculus:

- gradient: measures the rate and direction of change in a scalar field; the gradient of a scalar field is a vector field.
- curl: measures a vector field's tendency to rotate about a point; the curl of a vector field is another vector field.
- divergence: measures a vector field's tendency to originate from or converge upon a given point.

A fourth operation, the Laplacian, is combination of the divergence and gradient operations.

Likewise, there are three important theorems related to these operators:

- Gradient theorem
- Stokes' theorem
- divergence theorem

Most of the analytic results are easily understood, in a more general form, using the machinery of differential geometry, of which vector calculus forms a subset.

## See also

es:Cálculo vectorial fr:Analyse vectorielle it:Calcolo vettoriale he:אנליזה וקטורית ja:ベクトル解析 pt:Cálculo vetorial zh:向量分析