Vector Calculus

Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow.

Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra, which uses exterior products does generalize, as discussed below.

Read more about Vector Calculus:  Basic Objects, Theorems

Other articles related to "vector calculus, vector, vectors":

Disintegration Theorem - Applications - Vector Calculus
... can also be seen as justifying the use of a "restricted" measure in vector calculus ... For instance, in Stokes' theorem as applied to a vector field flowing through a compact surface Σ ⊂ R³, it is implicit that the "correct" measure on Σ is the disintegration of three-dimensional ...
Line Integral - Vector Calculus - Applications
... on a particle traveling on a curve C inside a force field represented as a vector field F is the line integral of F on C ...
Vector Calculus - Generalizations - Other Dimensions
... in a more general form, using the machinery of differential geometry, of which vector calculus forms a subset ... of view, the various fields in (3-dimensional) vector calculus are uniformly seen as being k-vector fields scalar fields are 0-vector fields, vector fields are 1-vector fields ... In higher dimensions there are additional types of fields (scalar/vector/pseudovector/pseudoscalar corresponding to 0/1/n−1/n dimensions, which is exhaustive in dimension 3), so one ...
Alternative Formulations of Maxwell's Equations
... Homogeneous equations Nonhomogeneous equations Vector calculus (fields) Vector calculus (potentials, any gauge) identities QED, vector calculus (potent ... (also called scalar potential) φ, and the magnetic potential A, (also called vector potential) ... even with just a glance at the equations—using covariant and contravariant four-vectors and tensors ...

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