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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

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 the Jacobian determinant.

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

**Formula of the 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**