Square 2 × 2 Jacobian
For (mathbf F=(x^2y,\sin x+y^3)), the rows come from the two components and the columns from (x,y).
(J_{\mathbf F}=\begin{bmatrix}2xy&x^2\\\cos x&3y^2\end{bmatrix}).
Jacobian Matrix Calculator with Steps
Enter each component of a vector-valued function on its own line. The calculator differentiates every component with respect to every variable, builds the Jacobian, and can evaluate the matrix at a point—all locally in your browser.
Enter the vector function
Use ^ for powers and * between variables. Functions include sin, cos, exp, ln, sqrt, and abs.
Use 1–6 distinct single-letter variables.
Enter one coordinate per variable in the same order, or leave blank for a symbolic Jacobian only.
Try an example
Keyboard shortcut: Ctrl/⌘ + Enter.
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Jacobian matrix
Symbolic Jacobian
The symbolic matrix will appear here.
Jacobian at the point
Enter an optional point to evaluate the matrix.
Step-by-step partial derivatives
Each row of partial derivatives will appear here.
Calculator methodology and limits
- Developed and reviewed by
- Starlight Robotics calculator team
- Last reviewed
- Method
- Symbolic first partial derivatives arranged with component functions as rows and input variables as columns.
- Input limits
- One to six functions, one to six distinct single-letter variables, and up to 300 characters per function.
- Privacy
- Functions, coordinates, and results stay in your browser; no solving API receives them.
- Domain note
- A symbolic entry can exist while evaluation fails at a point. Logarithms, roots, denominators, and absolute values can restrict differentiability or the real domain.
How to use the Jacobian calculator
- Enter the output components (f_1,f_2,\ldots,f_m), one per line. Their line order becomes the row order.
- List the input variables (x_1,x_2,\ldots,x_n). Their order becomes the column order.
- Optionally enter a point with exactly one coordinate per variable. Leave it blank if you only need the symbolic matrix.
- Select Calculate Jacobian, then inspect every partial derivative and the assembled (m\times n) matrix.
Jacobian matrix formula
For (mathbf F:\mathbb R^n\to\mathbb R^m), where (mathbf F=(f_1,\ldots,f_m)), the Jacobian is the (m\times n) matrix of first partial derivatives:
\[J_{\mathbf F}(\mathbf x)=\left[\frac{\partial f_i}{\partial x_j}\right]_{m\times n}=\begin{bmatrix}\frac{\partial f_1}{\partial x_1}&\cdots&\frac{\partial f_1}{\partial x_n}\\\vdots&\ddots&\vdots\\\frac{\partial f_m}{\partial x_1}&\cdots&\frac{\partial f_m}{\partial x_n}\end{bmatrix}\]
Each partial derivative treats all other input variables as constants, following the standard definition in OpenStax Calculus, Volume 3, section 4.3. Evaluating the entries at a point gives the best linear map there only when the function is differentiable at that point.
Worked examples
Rectangular 3 × 2 Jacobian
A map from two inputs to three outputs has three rows and two columns. Jacobian matrices do not need to be square.
The dimensions always equal outputs × inputs.
Scalar output
For one component (f(x,y)), the Jacobian is the row matrix ([f_x\;f_y]).
The gradient contains the same entries but is conventionally written as a column vector.
Jacobian matrix FAQs
What is a Jacobian matrix?
It is the matrix of all first partial derivatives of a vector-valued function. Entry ((i,j)) is ∂(f_i)/∂(x_j).
How are the rows and columns ordered?
Rows follow the component functions from top to bottom. Columns follow the variables from left to right exactly as entered.
Does a Jacobian have to be square?
No. A function with (n) inputs and (m) outputs has an (m\times n) Jacobian. It is square only when the input and output dimensions match.
Is the gradient the same as the Jacobian?
For a scalar-valued function, the Jacobian is a one-row matrix containing the same partial derivatives. The gradient is usually the transpose: a column vector.
What is the difference between a Jacobian matrix and a Jacobian determinant?
The matrix contains the partial derivatives. The Jacobian determinant is the determinant of that matrix and exists only for a square Jacobian.
Can the calculator evaluate the matrix at a point?
Yes. Enter finite coordinates in variable order. Leave the point blank to calculate the symbolic matrix without numerical evaluation.
Does an evaluated Jacobian always describe a derivative?
Not necessarily. Existing partial derivatives alone do not guarantee differentiability. Corners, discontinuities, and restricted domains still require mathematical judgment.
Does my input leave the browser?
No. Parsing, symbolic differentiation, matrix assembly, and optional point evaluation run on your device.