What is a Jacobian matrix?

The Jacobian matrix of a vector-valued function collects all first-order partial derivatives. For a system of m functions and n variables, it is an m×n matrix where entry (i,j) is the partial derivative of the i-th function with respect to the j-th variable.

What is the Jacobian determinant used for?

The determinant of the Jacobian matrix (when it is square) measures how a transformation scales areas or volumes locally. It appears in the change-of-variables formula for integrals and indicates whether a transformation is locally invertible at a point.

How are partial derivatives computed here?

This tool uses high-accuracy numerical differentiation (symmetric finite differences with a small step h) to approximate each partial derivative at the evaluation point you specify. Results match symbolic derivatives to 5–6 significant figures for smooth functions.

What function syntax is supported?

Use standard JavaScript math expressions: x^2 or x**2 for powers, sin(x), cos(x), tan(x), exp(x), log(x) for natural log, sqrt(x), abs(x). Multiplication must be explicit: write 2*x not 2x.

Why might the determinant be zero?

A zero Jacobian determinant at a point means the transformation is not locally invertible there — the functions are dependent in a neighborhood of that point. This is important for identifying critical points in optimization and checking if a coordinate change is valid.

Jacobian Calculator

Name: Jacobian Calculator
Availability: InStock
Author: Nham Vu

Enter 2 or 3 functions of x, y (and z) to compute the Jacobian matrix and its determinant instantly.

Number of variables

2 variables (x, y) 3 variables (x, y, z)

Functions

f(·) =

g(·) =

Evaluation point

x =

y =

Syntax reference

x^2 or x**2power sin(x), cos(x)trig exp(x)e^x log(x)natural log sqrt(x)square root abs(x)absolute value 2*x*yexplicit multiply

Enter functions and click Compute Jacobian

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Summary

Enter 2 or 3 functions of x, y (and z) to compute the Jacobian matrix and its determinant instantly.

How it works

Choose 2-variable mode (functions of x, y) or 3-variable mode (functions of x, y, z).
Enter each function using standard math notation: x^2, sin(x), exp(y), log(x), etc.
Click "Compute Jacobian" to evaluate partial derivatives symbolically via numeric differentiation.
The Jacobian matrix is displayed with each partial derivative entry.
The determinant (Jacobian determinant) is computed and shown below the matrix.

Use cases

Verify Jacobian matrices for multivariable calculus homework.
Compute the Jacobian determinant needed for change-of-variables in double/triple integrals.
Check partial derivative computations for coordinate transformations (polar, cylindrical, spherical).
Analyze local invertibility of a vector-valued function at a point.
Support robotics or physics calculations involving Jacobians of kinematic functions.
Cross-check results when working through vector calculus problems.

Frequently Asked Questions

Last updated: 2026-07-23 · Reviewed by Nham Vu