Jacobian

For a function , the derivative of the function can be viewed as an matrix that takes all the partial derivatives:

This matrix is often called the Jacobian matrix. The matrix can be seen as transforming vectors of rates of change of inputs to vectors of rates of change of outputs.

When the Jacobian matrix is a square matrix, the determinant of it is often referred to as Jacobian determinant. Both the matrix and the determinant are often referred to simply as the Jacobian in literature. ¹ Jacobian determinant plays an important role in measuring volume change under transformations.

Algebraic Interpretations

Each row of the Jacobian matrix is the gradient of one of the output component of the multi-valued function:

For example, given a function

The Jacobian matrix is

See Also

• divergence
• curl
• the derivative of the function
• Hessian
• linear approximation - the Jacobian matrix represent the best linear approximation of the function in a neighborhood of

Footnotes

1. Jacobian matrix and determinant - Wikipedia ↩