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In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors.The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
The Jacobi triple product identity is the Macdonald identity for the affine root system of type A 1, ... Jacobi's proof relies on Euler's pentagonal number theorem, ...
Suppose a function f(x, y, z) = 0, where x, y, and z are functions of each other. Write the total differentials of the variables = + = + Substitute dy into dx = [() + ()] + By using the chain rule one can show the coefficient of dx on the right hand side is equal to one, thus the coefficient of dz must be zero () + = Subtracting the second term and multiplying by its inverse gives the triple ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
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:
The vector triple product is the cross product of a vector with the result of another cross product, ... Proof by substitution. Evaluation of the cross product gives
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.
Proof: Both sides change signs upon switching two indices, ... If c = (c 1, c 2, c 3) is a third vector, then the triple scalar product equals ...