Question

State the definition of the Jacobian.

Short answer

The Jacobian is the matrix of all first-order partial derivatives of a vector-valued function.

Step-by-step solution

Definition

The Jacobian matrix, often simply called the Jacobian, is the matrix of all first-order partial derivatives of a vector-valued function.

Formulation

Suppose that \(f: R^n \rightarrow R^m\) is a function that maps a vector in R^n to R^m. If all the first-order partial derivatives of \(f\) exist, the Jacobian of \(f\) is an m x n matrix, often denoted by \(J(f)\) or \(Df\), defined by: \[ J(f) = \[ \[ \frac{\partial f1}{\partial x1}, \frac{\partial f1}{\partial x2}, ..., \frac{\partial f1}{\partial xn} \], \[ \frac{\partial f2}{\partial x1}, \frac{\partial f2}{\partial x2}, ..., \frac{\partial f2}{\partial xn} \], ..., \[ \frac{\partial fm}{\partial x1}, \frac{\partial fm}{\partial x2}, ..., \frac{\partial fm}{\partial xn} \] \] \]

Explanation

Each entry in this matrix is a first-order partial derivative. The rows correspond to the functions \(f1, f2, ..., fm\) and the columns correspond to the variables \(x1, x2, ..., xn\). Thus, the entry in the i-th row and j-th column of the Jacobian matrix is the partial derivative of the i-th function with respect to the j-th variable.