# Jacobian matrix

This article describes an analogue for functions of multiple variables of the following term/fact/notion for functions of one variable: derivative

## Importance

The Jacobian matrix is the appropriate notion of **derivative** for a function that has multiple inputs (or equivalently, vector-valued inputs) and multiple outputs (or equivalently, vector-valued outputs).

## Definition at a point

### Direct epsilon-delta definition

### Definition at a point in terms of gradient vectors as row vectors

Suppose is a vector-valued function with -dimensional inputs and -dimensional outputs. Explicitly, suppose is a function with inputs and outputs . Suppose is a point in the domain of such that is differentiable at for . Then, the **Jacobian matrix** of at is a matrix of *numbers* whose row is given by the gradient vector of at .

Explicitly, in terms of rows, it looks like:

### Definition at a point in terms of partial derivatives

Suppose is a vector-valued function with -dimensional inputs and -dimensional outputs. Explicitly, suppose is a function with inputs and outputs . Suppose is a point in the domain of such that is differentiable at for . Then, the **Jacobian matrix** of at is a matrix of *numbers* whose entry is given by:

Here's how the matrix looks:

Note that for this definition to be correct, it is still necessary that the gradient vectors exist. If the gradient vectors do not exist but the partial derivatives do, a matrix can still be constructed using this recipe but it may not satisfy the nice behavior that the Jacobian matrix does.

## Definition as a function

### Definition in terms of gradient vectors as row vectors

Suppose is a vector-valued function with -dimensional inputs and -dimensional outputs. Explicitly, suppose is a function with inputs and outputs . Then, the **Jacobian matrix** of is a matrix of *functions* whose row is given by the gradient vector of . Explicitly, it looks like this:

Note that the domain of this function is the set of points at which all the s individually are differentiable.

### Definition in terms of partial derivatives

Suppose is a vector-valued function with -dimensional inputs and -dimensional outputs. Explicitly, suppose is a function with inputs and outputs . Then, the **Jacobian matrix** of is a matrix of *functions* whose entry is given by:

wherever all the s individually are differentiable in the sense of the gradient vectors existing. Here's how the matrix looks:

If the gradient vectors do not exist but the partial derivatives do, a matrix can still be constructed using this recipe but it may not satisfy the nice behavior that the Jacobian matrix does.

## Particular cases

Case | What happens in that case? |
---|---|

is a real-valued function of one variable. The Jacobian matrix is a matrix whose entry is the ordinary derivative. | |

, | is a vector-valued function of one variable. We can think of it as a parametric curve in . The Jacobian matrix is a matrix which, read as a column vector, is the parametric derivative of the vector-valued function. |

, | is a real-valued function of multiple variables. The Jacobian matrix is a matrix which, read as a row vector, is the gradient vector function. |

is a linear or affine map. | The Jacobian matrix is the same as the matrix describing (or, if is affine, the matrix describing the linear part of ). |

, and we are identifying the spaces of inputs and outputs of . | The Jacobian matrix can then be thought of as a linear self-map from the -dimensional space to itself. In this context, we can consider the Jacobian determinant. |