# Jacobian Calculator

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## Jacobian Determinant Calculator

Find the Jacobian matrix and determinant with this calculator. You can use functions containing 2 or 3 variables and get their step-wise computation.

## How to use this tool?

The Jacobian matrix calculator requires three steps like most determinant calculators for multi-variables.

1. Choose the number of the variables.
2. Enter the values.
3. Click on the “Calculate”.

## What is the Jacobian matrix?

In a multivariable setting, the Jacobian matrix represents the partial derivatives of a set of functions. If you have a vector function $$F= (f_1,f_2, … f_n)$$ mapping from $$\mathbb{R}^n$$ to $$\mathbb{R}^n$$, the Jacobian matrix is a square matrix of order n containing all first-order partial derivatives of these functions.

Formally, the Jacobian matrix J is defined as:

$J = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1} & \frac{\partial f_n}{\partial x_2} & \cdots & \frac{\partial f_n}{\partial x_n} \end{pmatrix}$

Each row in this matrix corresponds to one of the outputs (like $$u$$ or $$v$$, i.e., $$f_1$$ and $$f_2$$), and each column corresponds to one of the inputs (like $$x$$ or $$y$$).

## What is the Jacobian Determinant?

In the context of the Jacobian matrix, the determinant (called the Jacobian determinant) tells us two critical things:

• How areas/volumes change: When you apply the function to a small area (or volume) around a point, the Jacobian determinant tells you how much that area (or volume) is stretched or shrunk.
• Whether the function is invertible locally: If the Jacobian determinant is not zero at a point, it means that the function can be reversed (inverted) around that point.

### Example:

Think of a flat rubber sheet representing your input space (x and y). A small

square is drawn on this sheet. Now, imagine this sheet is deformed somehow

(like twisting, stretching, or compressing), representing your function.The

drawn square might turn into a different shape (like a rectangle,

parallelogram,or some other distorted shape).

• The Jacobian matrix tells exactly how the sheet is being twisted or stretched at any point.
• The Jacobian determinant gives a single number representing how the area of your square changes — does it get larger, smaller, or stay the same size? And does the deformation flip the square (changing its orientation)?

## How to find the Jacobian Matrix and Determinant?

Let’s break down the method for 2 and 3 variables to find the Jacobian matrix and its determinant.

Step 1: Define the Function

For Two Variables: Consider a function $F(x,y) = (f_1(x,y),f_2(x,y))$ For Three Variables: Consider a function $G(x,y,z) =(g_1(x,y,z),g_2(x,y,z),g_3(x,y,z))$

Step 2: Compute Partial Derivatives

Partial Derivatives: For each component function $$f_1, f_2$$ in the two-variable case or $$g_1, g_2, g_3$$ in the three-variable case), compute the partial derivatives with respect to each variable.

Step 3: Form the Jacobian Matrix.

For Two Variables: The Jacobian matrix $$J_F$$ is a 2x2 matrix where each entry is a partial derivative:

$$J = \begin{pmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{pmatrix}$$

For Three Variables: The Jacobian matrix $$J_G$$ is a 3x3 matrix where each entry is a partial derivative:

$$J = \begin{pmatrix} \frac{\partial g_1}{\partial x} & \frac{\partial g_1}{\partial y} & \frac{\partial g_1}{\partial z} \\ \frac{\partial g_2}{\partial x} & \frac{\partial g_2}{\partial y} & \frac{\partial g_2}{\partial z} \\ \frac{\partial g_3}{\partial x} & \frac{\partial g_3}{\partial y} & \frac{\partial g_3}{\partial z} \end{pmatrix}$$

Step 4: Calculate the Determinant.

Determinant Calculation: Calculate the determinant of the Jacobian matrix, which involves algebraic manipulation based on the matrix size.

### Example (For 2 variables):

Function Definition

Consider the function

$$\mathbf{F}(x, y) = (x^2 + y, e^x + y^2)$$

Step-by-Step Calculation

1. Calculate Partial Derivatives
• For $$f_1 (x,y) = x^2 + y$$

\begin{align*} \frac{\partial f_1}{\partial x} &= 2x, \frac{\partial f_1}{\partial y} &= 1 \end{align*}

• For $$f_2 (x,y) = e^x + y^2$$

$\frac{\partial f_2}{\partial x} = e^x,\quad \frac{\partial f_2}{\partial y} = 2y$

1. Form the Jacobian Matrix

$$J_{\mathbf{F}} = \begin{pmatrix} 2x & 1 \\ e^x & 2y \end{pmatrix}$$

1. Calculate the Determinant

$$\det(J_{\mathbf{F}}) = (2x)(2y) - (1)(e^x) = 4xy - e^x$$

### Example (For 3 variables):

Function Definition

$\mathbf{G}(x, y, z) = (xyz, x^2 + z, y - e^z)$

Step-by-Step Calculation

1. Calculate Partial Derivatives
• For $g_1 (x,y,z) = xyz$

$\frac{\partial g_1}{\partial x} = yz, \quad \frac{\partial g_1}{\partial y} = xz, \quad \frac{\partial g_1}{\partial z} = xy$

• For $g_2 (x,y,z) = x^2 + z$

$\frac{\partial g_2}{\partial x} = 2x, \quad \frac{\partial g_2}{\partial y} = 0, \quad \frac{\partial g_2}{\partial z} = 1$

• For $g_3 (x,y,z) = y - e^z$

$\frac{\partial g_3}{\partial x} = 0, \quad \frac{\partial g_3}{\partial y} = 1, \quad \frac{\partial g_3}{\partial z} = -e^z$

1. Form the Jacobian Matrix

$J_{\mathbf{G}} = \begin{pmatrix} yz & xz & xy \\ 2x & 0 & 1 \\ 0 & 1 & -e^z \end{pmatrix}$

1. Calculate the Determinant

To calculate the determinant of the 3x3 Jacobian matrix $$J_G$$, you expand along a row or column. Typically, the first row is a convenient choice. The expansion is given by:

$\det(J_{\mathbf{G}}) = yz \cdot \det\left(\begin{pmatrix} 2x & 0 \\ 0 & 1 \end{pmatrix}\right)$
$$\begin{pmatrix} 0 & 1 \\ 1 & -e^z \end{pmatrix} - xz \cdot \det \begin{pmatrix} 2x & 1 \\ 0 & -e^z \end{pmatrix} + xy \cdot \det \begin{pmatrix} 2x & 0 \\ 0 & 1 \end{pmatrix}$$

The determinants of the 2x2 matrices are calculated as:

\begin{align*} \det \begin{pmatrix} 0 & 1 \\ 1 & -e^z \end{pmatrix} &= (0)(-e^z) - (1)(1) = -1 \\ \det \begin{pmatrix} 2x & 1 \\ 0 & -e^z \end{pmatrix} &= (2x)(-e^z) - (1)(0) = -2xe^z \\ \det \begin{pmatrix} 2x & 0 \\ 0 & 1 \end{pmatrix} &= (2x)(1) - (0)(0) = 2x \\ \end{align*}

So, the complete determinant is:

$\det(J_{\mathbf{G}}) = yz(-1) - xz(-2xe^z) + xy(2x) = -yz + 2x^2ze^z + 2x^2y$

## Significance:

Local Linear Approximation: The Jacobian matrix is used to approximate the behavior of a function near a point. If the Jacobian determinant at a point is non-zero, the function is locally invertible near that point.

Volume Change: In geometric terms, the Jacobian determinant at a point gives the factor by which the function F scales volumes near that point. If it's positive, the orientation is preserved; if negative, the orientation is reversed.

Coordinate Transformations: In integration, the Jacobian determinant is used to change variables. It tells you how much the volume element (like dx, dy in 2D) is stretched or shrunk under the transformation.