Jacobian Matrix Calculator

The Jacobian Matrix Calculator builds the Jacobian matrix — the full matrix of first-order partial derivatives — for a vector-valued function F = (f₁, f₂, …) with respect to variables (x₁, x₂, …). It shows the symbolic Jacobian, the symbolic determinant when the matrix is square, an invertibility verdict, a numeric evaluation at any point you choose, and a full step-by-step derivation of every entry.

It's built for multivariable calculus and linear algebra students checking a homework answer, and for anyone who needs a Jacobian without differentiating a whole grid of partial derivatives by hand.

How to use it

1. Enter your functions f₁, f₂, … — one per line, or separated by semicolons — using standard math notation: ^ for powers, * for multiplication, plus sqrt(), sin(), cos(), log(), and the other functions supported by mathjs's expression parser. Up to 4 functions.
2. List the variables to differentiate with respect to, comma-separated single letters (e.g. x, y, z). Up to 4 variables.
3. Optionally enter a point to evaluate the Jacobian numerically — one value per variable, comma-separated. Leave it blank to see the symbolic Jacobian only.
4. Read Jacobian Matrix J for the matrix of partial derivatives (row i, column j is ∂fᵢ/∂xⱼ), Shape for its dimensions, and Determinant |J| for the symbolic determinant when the matrix is square.
5. Evaluated at the Point shows the numeric Jacobian, the determinant's value, and whether the matrix is invertible there. Step-by-Step Derivation shows every partial derivative computed along the way.

Results update automatically as you type. For example, F = (x^2*y, x + z) with variables x, y, z gives a 2×3 Jacobian [[2xy, x², 0], [1, 0, 1]] — non-square, so no determinant — while F = (xy, x + y) with variables x, y gives the square Jacobian [[y, x], [1, 1]], determinant y − x, invertible everywhere except along the line y = x.

What is a Jacobian matrix?

For a vector-valued function F(x₁, …, xₙ) = (f₁, …, fₘ), the Jacobian is the m×n matrix:

J[i][j] = ∂fᵢ/∂xⱼ

Each row is one function's gradient; each column is how every function responds to one variable. It's the multivariable generalization of a single derivative, and it's what a first-order (linear) approximation of F near a point looks like.

Why does invertibility matter?

When the Jacobian is square (as many functions as variables) and its determinant is nonzero at a point, the inverse function theorem guarantees F is locally invertible there — a small change in the output can be traced back to a unique small change in the input. This calculator checks that condition automatically: if the determinant is identically 0, the matrix is singular everywhere (F is never locally invertible); otherwise it's invertible at any point where the determinant doesn't happen to vanish.

What functions can I enter?

Anything mathjs's expression evaluator understands: polynomials (x^2*y - 2*z), trigonometric functions (sin(x*z), cos(y)), roots and logarithms (sqrt(x), log(y)), and combinations of these. Angles in trig functions are in radians by default — write pi for π. A symbol you list as a function but not as a variable (or vice versa) is treated as a constant during differentiation.

To differentiate a single function with respect to one variable at a time, see the Partial Derivative Calculator; for 2×2/3×3 numeric matrix operations (determinant, inverse, multiplication), see the Matrix Calculator.

Privacy

This calculator runs entirely in your browser. Your functions, variables, and point are never uploaded, logged, or stored — the computation happens locally on your device.