# Explain the term Jacobian.

The term Jacobian is usually used in vector calculus. Let's see what it means.

## Answer: The jacobian matrix is a matrix of partial derivatives of a multi-valued function.

We will understand what Jacobian means.

**Explanation**:

Jacobian is used in vector calculus. We use the term jacobian to refer Jacobian matrix.

Let x = (x_{1}, x_{2}, ... , x_{n}).

We define function f: R^{n} **ā **R^{m} as f(x) = (f_{1}, f_{2}, ... , f_{m}).

The Jacobian matrix J(x_{1}, x_{2}, ..., x_{n}) is given by,

\(\begin{bmatrix}

\frac{\partial f_1 }{\partial x_1} & \frac{\partial f_1 }{\partial x_2} & ... & \frac{\partial f_1}{\partial x_n}\\

:& : &: & : \\

:& : &: & : \\

\frac{\partial f_m }{\partial x_1} & \frac{\partial f_m }{\partial x_2} & ... & \frac{\partial f_m}{\partial x_n}

\end{bmatrix}\)

For example, Jacobian of f(x, y) = (x + y, 2x) is given by

J(x, y) = \(\begin{bmatrix}

1 & 1 \\

2 & 0

\end{bmatrix}\)

### So, the jacobian matrix is a matrix of partial derivatives of a multi-valued function.