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# Jacobian Calculator

To find the Jacobian matrix, select variables, enter the functions in the required input boxes, and press the** calculate** button using Jacobian calculator

Table of Contents:

Jacobian calculator is used to find the Jacobian matrix & determinant after taking the derivative of the given function. This Jacobian matrix calculator finds the matrix for two and three variable functions.

## What is the Jacobian matrix?

In vector calculus, the Jacobian matrix of multivariable-variable functions is the matrix of all its 1st order partial derivatives.

The Jacobian matrix takes an equal number of rows and columns as an input i.e., 2x2, 3x3, and so on. In other words, the input values must be a square matrix. The determinant of the Jacobian matrix is referred to as Jacobian determinant.

It is denoted by **J** and the entry (i, j) such as **J _{i}_{,j} = ∂f_{i}/ ∂x_{j}**

### Formula of Jacobian matrix

Below is the general formula to find the Jacobian matrix.

## How to calculate the Jacobian matrix?

Below is a solved example of the Jacobian matrix.

**Example**

Find Jacobian matrix of x = x^{2} + 2y^{2} & y = 3x – 2y with respect to x&y.

**Solution **

**Step 1: **Write the given functions in a matrix.

x = x^{2} + 2y^{2}

y = 3x – 2y

**Step 2:** Find the partial derivative of column 1 w.r.t **“x”** and column 2 w.r.t **“y”**.

∂/∂x (x^{2}, 3x) = 2x, 3

∂/∂y (2y^{2}, -2y) = 4y, -2

**Step 3: **Write the terms in the matrix form.

This is the required 2x2 Jacobian matrix of the given functions.

The determinant of this matrix is -4x -12y

**Jacobian matrix = -4x – 12y**

### How to find the Jacobian of a 3x3 matrix

Find Jacobian matrix of x = 3x^{3} + 4y^{2} – z^{2}, y = 5x – 3y + 6z, and z = x + y + z with respect to x,y&z.

**Solution **

To find the 3x3 Jacobian matrix, follow the below steps.

**Step 1:** Write the given functions in a matrix.

x = 3x^{3} + 4y^{2} – z^{2}

y = 5x – 3y + 6z

z = x + y + z

**Step 2:** Find the partial derivative of column 1 w.r.t **“x”**, column 2 w.r.t **“y”**, and column 3 w.r.t **“z”**.

∂/∂x (3x^{3}, 5x, x) = 9x^{2}, 5, 1

∂/∂y (4y^{2}, -3y, y) = 8y, -3, 1

∂/∂z (z^{2}, 6z, z) = 2z, 6, 1

**Step 3:** Write the terms in the matrix form.

This is the required 3x3 Jacobian matrix of the given functions.

The determinant of this matrix is -81x^{2} + 8y – 16z

**Jacobian matrix = -81x ^{2} + 8y – 16z**