Jacobian Matrix Calculator

You enter the first-order partial derivatives of a vector valued function F, one value per cell, and the calculator assembles them into the 2x2 or 3x3 Jacobian matrix J, where entry J[i][j] is the partial derivative of component i with respect to variable j. It then returns the determinant, its absolute value, the trace, and the curl. All entries are evaluated at a single point, so you supply numbers rather than symbolic formulas.

Differentiate each output component of your function with respect to each input variable, then evaluate those derivatives at the point you care about. For F(x, y) = (x squared plus y, x times y) at the point (1, 3), the partials are 2x = 2, 1, y = 3, and x = 1, so you would enter 2, 1, 3, and 1. A computer algebra system can give you the symbolic partials if the derivatives are hard to do by hand.

The absolute value of the determinant is the local scaling factor for area (2x2) or volume (3x3) when you change variables in a multiple integral, so dx dy becomes |det J| du dv. A determinant of zero means the map is not locally invertible at that point: the inverse function theorem fails and the transformation collapses area or volume to zero there.

A multivariable Newton step solves J times delta equals minus F and updates the guess by delta, which requires the Jacobian to be invertible. If the determinant is zero, the linear system has no unique solution and the step cannot be taken, so the calculator flags a singular Jacobian. Near-zero determinants are also treated as singular because they make the step numerically unstable.

The trace is the sum of the diagonal entries, which equals the divergence of the vector field F, the sum of each function's partial derivative with respect to its own variable. The curl measures local rotation: for a 2x2 system it is the scalar (partial of f2 by x) minus (partial of f1 by y), and for a 3x3 system it is the standard three component curl vector. Both are read directly off the Jacobian entries.

No. It works with the numeric partial derivative values you supply, so the arithmetic for the determinant, trace, and curl is exact rather than a finite-difference approximation. If you need the symbolic partial derivatives first, compute them by hand or with a computer algebra tool, evaluate them at your point, and enter the resulting numbers here.