Search results
Results From The WOW.Com Content Network
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. [ 1 ]
The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant. The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
Vector calculus, a branch of mathematics concerned with differentiation and integration of vector fields; Vector differential, or del, a vector differential operator represented by the nabla symbol ; Vector Laplacian, the vector Laplace operator, denoted by , is a differential operator defined over a vector field; Vector notation, common ...
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
In vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, [1] [2] [3] / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space. [1] A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane.
In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve. For example, the line integral over a scalar field (rank 0 tensor) can be interpreted as the area under the field carved out by a particular curve.