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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 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 single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus: [1]: 543ff Gradient theorem; Stokes' theorem; Divergence theorem; Green's theorem.
In differential geometry, the four-gradient (or 4-gradient) is the four-vector analogue of the gradient from vector calculus. In special relativity and in quantum mechanics , the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors .
In the tensor calculus formulation, the electromagnetic tensor F αβ is an antisymmetric covariant order 2 tensor; the four-potential, A α, is a covariant vector; the current, J α, is a vector; the square brackets, [ ], denote antisymmetrization of indices; ∂ α is the partial derivative with respect to the coordinate, x α.
The first chapter offers a valuable review of vector calculus, which is essential for understanding this subject. While most other authors, including those aimed at a more advanced audience, denote the distance from the source point to the field point by | x − x ′ | {\displaystyle |\mathbf {x} -\mathbf {x} '|} , Griffiths uses a script r ...
In mathematical physics, in particular electromagnetism, the Riemann–Silberstein vector [1] or Weber vector [2] [3] named after Bernhard Riemann, Heinrich Martin Weber and Ludwik Silberstein, (or sometimes ambiguously called the "electromagnetic field") is a complex vector that combines the electric field E and the magnetic field B.
An example of a solenoidal vector field, (,) = (,) In vector calculus a solenoidal vector field (also known as an incompressible vector field , a divergence-free vector field , or a transverse vector field ) is a vector field v with divergence zero at all points in the field: ∇ ⋅ v = 0. {\displaystyle \nabla \cdot \mathbf {v} =0.}
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