Explain the term Jacobian.

Solution:

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

We will understand what Jacobian means.

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

Let x = (x₁, x₂, ... , x_n).

We define function f: Rⁿ ➝ R^m as f(x) = (f₁, f₂, ... , f_m).

The Jacobian matrix J(x₁, x₂, ..., 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.

Explain the term Jacobian.

Summary:

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