The Jacobian matrix of a function at the point with respect to some choice of bases for and is the matrix of the linear map from into that generalizes the definition of the derivative of a function on . It can be defined as the matrix of the linear map , such that
The linear map that satisfies the above limit is called the derivative of at . It is easy to show that the Jacobian matrix of a given differentiable function at with respect to chosen bases is just the matrix of http://planetmath.org/node/841partial derivatives of the component functions of at :
A more concise way of writing it is
where is the partial derivative with respect to the ’th variable and is the gradient of the ’th component of .
Given local coordinates for some real submanifold of , it is easy to show that the effect of a change of coordinates on volume forms is a local scaling of the volume form by the determinant of the Jacobian matrix of the derivative of the backwards change of coordinates, which is called the inverse Jacobian. The determinant of the inverse Jacobian is thus commonly seen in integration over a change of coordinates.
where , , and is the derivative of the function which maps points in to points in .
|Date of creation||2013-03-22 11:58:33|
|Last modified on||2013-03-22 11:58:33|
|Last modified by||PhysBrain (974)|
|Synonym||derivative of a multivariable function|
|Synonym||derivative of a vector-valued function|