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A Jacobian matrix, sometimes simply called a Jacobian, is a matrix of first order partial derivatives (in some cases, the term "Jacobian" also refers to the determinant of the Jacobian matrix).

For a function , the Jacobian is the following matrix:

or, in Einstein notation,

Note that in some conventions, the Jacobian is the transpose of the above matrix.

Jacobians where are square matrices, and are commonly used when changing coordinates, especially when taking multiple integrals and determining whether complex functions are holomorphic. For example, a Jacobian representing a change in variables from to and to in two dimensions is represented as

A Jacobian matrix is what is usually meant by the derivative of higher-dimensional functions; indeed, differentiability in the components of a Jacobian guarantees differentiability in the function itself. In the case of a multivariable function , the Jacobian matrix with respect to the input variables is simply the gradient of the function. The Jacobian is also related to the Hessian matrix by

Applications

Jacobian matrices are useful in integration when changing coordinate systems. For example, given a two dimensional coordinate transformation, the double integral of becomes

When working with one independent variable, this becomes

which, when used to compute an integral, yields the formula known as integration by substitution.

Examples

Jacobian matrices are useful in integration when changing coordinate systems. For example, given a two dimensional coordinate transformation, the double integral of becomes

When working with one independent variable, this becomes

which, when used to compute an integral, yields the formula known as integration by substitution. Find the area of a circle of radius by transforming from Cartesian coordinates to Polar coordinates.

Since and , the Jacobian determinant becomes

The integral now becomes

Since it is now in polar coordinates, we can add the bounds as

Now we can integrate it.

Which is to be expected. We have also found that the differential of is .

This same method can be used to find the volume of a sphere in spherical coordinates. Since

the Jacobian determinant evaluates to:

This means that for any function in spherical coordinates, the volume element is

The integral is therefore

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