A generalization of the derivative for vector-valued functions.
Jacobian matrix for f:Rn→Rm is defined as:
Jpf(x)=∂x1∂f1∂x1∂f2⋮∂x1∂fm∂x2∂f1∂x2∂f2⋮∂x2∂fm⋯⋯⋱⋯∂xn∂f1∂xn∂f2⋮∂xn∂fm
Jacobian of the transformation T:x=x(u,v),y=(u,v) is:
∂(u,v)∂(x,y)=det[∂u∂x∂u∂y∂v∂x∂v∂y]=∂u∂x∂v∂y−∂v∂x∂u∂y
Approximation
First order approximation of f(x) around x0 can be obtained by the Jacobian matrix.
A(x)=f(x0)+JpTf(x0)(x−x0)