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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.
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:
Jerrold Eldon Marsden (August 17, 1942 – September 21, 2010) was a Canadian mathematician. He was the Carl F. Braun Professor of Engineering and Control & Dynamical Systems at the California Institute of Technology . [ 1 ]
Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
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 .
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in ) bounded by C. It is the two-dimensional special case of Stokes' theorem (surface in R 3 {\displaystyle \mathbb {R} ^{3}} ).
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 vector calculus, a conservative vector field is a vector field that is the gradient of some function. [1] A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the ...