Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 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.

• Stokes' theorem
• divergence theorem

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

History

Quaternions were discovered by William Rowan Hamilton of Ireland in 1843. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a plane) to higher spatial dimensions. Quaternions are made of a 3 dimensional vector plus a scalar. Oliver Heaviside and Willard Gibbs among others developed vector algebra and vector calculus.

Some of Hamilton's supporters vociferously opposed the growing fields of vector algebra and vector calculus (developed by Oliver Heaviside and Willard Gibbs among others), maintaining that quaternions provided a superior notation. While this is debatable in three dimensions, quaternions cannot be used in other dimensions (though extensions like octonions and Clifford algebras may be more applicable). Vector notation nearly universally replaced quaternions in science and engineering by the mid-20th century.

See also: list of multivariable calculus topics.