## Introduction to Jacobian Calculator:

Jacobian Calculator is an online tool that is used to **find the multivariable** set of functions using the determinant matrix method. It helps in computing the several variable functions by taking their partial derivative with respect to its variable to analyze the small change in the given function system.

Our jacobian matrix calculator is very efficient as it provides you a solution of a jacobian method even for the complex variable function without taking any assistance from you except it takes the input value from you.

## What is Jacobian?

Jacobian method is a **numerical analysis** process which uses a partial differential method first order for solving several variable functions by taking the determinant matrix to get a solution. It is represented with the symbol J.

It is used in various field like in physic,scientific research work, data analysis or robotic programing to analyze the change in the behavior of a given multivariable function.

## Formula of Jacobian:

The **formula** of the Jacobian matrix depends on the multivariable function F(f1,f2,...,fn) and its partial derivative with respect to its variable term.Here is the Jacobian matrix of multivariable function of n by n order. The jacobian formula used by the jacobian calculator is,

$$ J \;=\; \left[ \begin{matrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \\ \end{matrix} \right] $$

## How to Calculate Jacobian?

To **calculate the Jacobian** matrix, the jacobian transformation calculator follows the below process in which it involves multivariable functions and compute the partial derivatives, add it into the Jacobian matrix. Here’s a detailed guide about its calculation process.

**Step 1**:

Identify the vector-valued function F(x) as F (f1,f2,f3,..,fn) for jacobian matrix.

**Step 2**:

Compute the partial derivatives for each function f_i with respect to its variable xi.

**Step 3**:

Construct the Jacobian Matrix J by adding the partial derivative function value in it.

$$ J \;=\; \left[ \begin{matrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \\ \end{matrix} \right] $$

**Step 4**:

Now take the determinant of a given jacobian matrix to find the change in the several variable function solution.

## Solved Example of Jacobian:

The Jacobian calculator is helpful in getting the accurate answers but it is also important to understand the step by step calculation process so an **example** is given below,

### Example: Find the function using jacobian method

$$ f(x,y,z) \;=\; \left[ \begin{matrix} x^2\; +\; y\;+\; 2z \\ x^3\; +\; y \;+\; 3z \\ x^2+\; + \; y^2 \\ \end{matrix} \right] $$

**Solution**:

Determine the given multivariable function F=(f1,f2,f3,...,fn),

$$ f_1(x,y,z) \;=\; x^2 + y + 2z $$

$$ f_2(x,y,z) \;=\; x^3 + y + 3z $$

$$ f_3(x,y,z) \;=\; x^2 + y^2 $$

As the function has three variable so we make 3 by 3 jacobian matrix,

$$ J_f(x,y,z) \;=\; \left[ \begin{matrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} & \frac{\partial f_1}{\partial z} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} & \frac{\partial f_2}{\partial z} \\ \frac{\partial f_3}{\partial x} & \frac{\partial f_3}{\partial y} & \frac{\partial f_3}{\partial z} \\ \end{matrix} \right] $$

Take the partial differential for each variable in the several variable function

$$ \frac{\partial f_1}{\partial x} \;=\; 2x $$

$$ \frac{\partial f_1}{\partial y} \;=\; 1 $$

$$ \frac{\partial f_1}{\partial z} \;=\; 2 $$

$$ \frac{\partial f_2}{\partial x} \;=\; 3x $$

$$ \frac{\partial f_2}{\partial y} \;=\; 1 $$

$$ \frac{\partial f_2}{\partial z} \;=\; 3 $$

$$ \frac{\partial f_3}{\partial x} \;=\; 2x $$

$$ \frac{\partial f_3}{\partial y} \;=\; 2y $$

$$ \frac{\partial f_3}{\partial z} \;=\; 0 $$

Put these value in the above jacobian matrix.

$$ J_f(x,y,z) \;=\; \left[ \begin{matrix} 2x & 1 & 2 \\ 3x^2 & 1 & 3 \\ 2x & 2y & 0 \\ \end{matrix} \right] $$

Now this jacobian become a determinant matrix as,

$$ det(J_f) \;=\; \left| \begin{matrix} 2x & 1 & 2 \\ 3x^2 & 1 & 3 \\ 2x & 2y & 0 \end{matrix} \right| $$

Solve it to get solution, the determinant jacobian matrix solution is,

$$ Jacobian\; Determinant \;=\; 12x^2 y - 12xy + 2x $$

## How to Use Jacobian Calculator?

The jacobian matrix calculator has a simple design, so everyone can use it to calculate the function of the Jacobian matrix. Before adding the input value, you must follow some instructions. These instructions are:

- Enter the value of the given multivariable function f(x) for jacobian matrix solution in the input box.
- Select the number of order for variable function in jacobian transformation calculator.
- Review your input value before hitting the calculate button to start the calculation process to find the solution using the jacobian matrix.
- Click on the “
**Calculate**” button to get the required result for the jacobian several variable function problems. - If you want to try out our jacobian determinant calculator for the first time, then you can use the load example to see the calculation process of it.
- Click on the “Recalculate” button to get a new page for solving more Jacobian problem for several function.

## Outcome the Jacobian Matrix Calculator:

The jacobian integral calculator gives you the **solution** to a given multivariable function problem when you input the value into it. It may be included as:

**Result Option**:

You can click on the result option, it will provide you with a solution for the jacobian several variable function questions.

**Possible Step**:

The Possible Steps option of Jacobian Calculator provides you with a solution in which evaluation steps for a multivariable function of jocabianl questions.

## Advantages of Using Jacobian Transformation Calculator:

The jacobian method calculator has many valuable **advantages** that help you to get the solution of the multivariable function whenever you use it for calculation. These advantages are:

- Our jacobian integral calculator is a free tool that you can use to find the small change in the function in the Jacobian matrix question without paying any charge.
- It saves your time and effort from doing lengthy and complex calculations of the jacobian matrix question in less than a minute.
- The jacobian determinant calculator is an easy-to-use tool, so you do not need any technical expertise. to use it to calculate the given jacobian matrix function problems easily.
- Jacobian matrix calculator is a reliable tool that provides you with accurate solutions every time whenever you use it to calculate the multivariable jacobian function problems.
- Jacobian Calculator provides you with a solution procedure in a step-by-step method for more clarity.