A function f=f(x) of one variable is differentiable at x with derivative File:Hepa img514.gif if File:Hepa img515.gif

File:Hepa img516.gif

This definition can be generalized to the case of m functions of n variables. Then x and h are File:Hepa img517.gif matrices (n-vectors), f and R are File:Hepa img518.gif matrices, and one defines for example

File:Hepa img519.gif

File:Hepa img514.gif then becomes an  matrix, called the Jacobi matrix whose elements are the partial derivatives:

File:Hepa img520.gif

Other possible notations for File:Hepa img514.gif are:

File:Hepa img521.gif

The chain rule is valid in its usual form. If File:Hepa img522.gif then File:Hepa img523.gif . Note that this is a matrix product, and therefore non-commutative except in special cases. In terms of matrix elements,

File:Hepa img524.gif

A coordinate transformation File:Hepa img525.gif is an important special case, with p=n, and with u=u(x) the inverse transformation of x=x(u). That is, u=u(x) = u(x(u)), and by the chain rule

File:Hepa img526.gif

i.e., the product of File:Hepa img527.gif and File:Hepa img528.gif is the unit matrix, or File:Hepa img529.gif .

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