Vector calculus

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Topics in calculus

Fundamental theorem | Function | Limits of functions | Continuity | Mean value theorem | Vector calculus | Tensor calculus


Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem | Related rates


Integration by substitution | Integration by parts | Integration by trigonometric substitution | Integration by disks | Integration by cylindrical shells | Improper integrals | Lists of integrals

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:

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