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.
- Choose the number of the variables.
- Enter the values.
- 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
- 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 \(f_2 (x,y) = e^x + y^2\)
\[\frac{\partial f_2}{\partial x} = e^x,\quad \frac{\partial f_2}{\partial y} = 2y\]
- Form the Jacobian Matrix
\(J_{\mathbf{F}} = \begin{pmatrix} 2x & 1 \\ e^x & 2y \end{pmatrix}\)
- 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
- 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 \[g_3 (x,y,z) = y - e^z\]
\[\frac{\partial g_3}{\partial x} = 0, \quad \frac{\partial g_3}{\partial y} = 1, \quad \frac{\partial g_3}{\partial z} = -e^z\]
- Form the Jacobian Matrix
\[J_{\mathbf{G}} = \begin{pmatrix} yz & xz & xy \\ 2x & 0 & 1 \\ 0 & 1 & -e^z \end{pmatrix}\]
- 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.