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The Jacobian matrix of functions \(u(x, y), v(x, y)\) holds all their first partial derivatives: row 1 is \(u_x, u_y\) and row 2 is \(v_x, v_y\). Its determinant, the Jacobian determinant, tells you how much a change of variables stretches area. For polar coordinates \(x = r\cos\theta,\ y = r\sin\theta\), the Jacobian determinant is \(r\), which is why \(dx\,dy = r\,dr\,d\theta\).
The Jacobian Matrix
One row per function, one column per variable. Each entry is a partial derivative. With three functions of three variables, it becomes a 3×3 matrix.
Example: \(u = x^2y,\ v = 5x + \sin y\)
\[J = \begin{bmatrix} 2xy & x^2 \\ 5 & \cos y \end{bmatrix}, \qquad \det J = 2xy\cos y – 5x^2.\]
At the point \((1, 0)\), \(J = \begin{bmatrix} 0 & 1 \\ 5 & 1 \end{bmatrix}\) and \(\det J = -5\). Type 1, 0 in the “At point” box to get these numbers.
Jacobians You Will See Again and Again
| Coordinates | Change of variables | Jacobian determinant |
|---|---|---|
| Polar | \(x = r\cos\theta,\ y = r\sin\theta\) | \(r\) |
| Cylindrical | \(x = r\cos\theta,\ y = r\sin\theta,\ z = z\) | \(r\) |
| Spherical | \(x = \rho\sin\phi\cos\theta,\ y = \rho\sin\phi\sin\theta,\ z = \rho\cos\phi\) | \(\rho^2\sin\phi\) |
| Linear | \(x = u + v,\ y = u – v\) | \(-2\) (use \(|{-2}| = 2\)) |
Use the example buttons above to see the full derivation of each one. Type Greek letters as words: theta, phi, rho.
Why the Jacobian Matters: Changing Variables in Integrals
When you switch an integral to new variables, the little area piece \(dx\,dy\) changes size. The Jacobian determinant is the conversion factor:
\[\iint_R f(x, y)\,dx\,dy = \iint_S f\big(x(u, v), y(u, v)\big)\,\left|\frac{\partial(x, y)}{\partial(u, v)}\right|\,du\,dv.\]
This is how a hard double integral over a circle becomes an easy one in polar coordinates, and how a triple integral over a ball becomes easy in spherical coordinates.
It is the 2D version of u-substitution. In one variable, \(dx = g'(u)\,du\). The Jacobian determinant plays the role of \(g'(u)\) when there are two or three variables. See the u-substitution guide for the one-variable idea.
Other Uses
- Linear approximation: near a point, a vector function behaves like its Jacobian matrix, just as a one-variable function behaves like its tangent line.
- Invertibility: if \(\det J \ne 0\) at a point, the transformation can be undone near that point (the inverse function theorem).
- Robotics and physics: Jacobians link joint speeds to the speed of a robot arm’s hand, and appear in fluid flow and thermodynamics.
Learn the Math Behind the Jacobian Calculator
The Jacobian collects all first partial derivatives. Its determinant is the stretch factor when you change variables in a double or triple integral.
Frequently Asked Questions
Is the Jacobian the matrix or the determinant?
Both names are used. “Jacobian matrix” means the matrix of partial derivatives. “Jacobian” alone often means its determinant, especially in a change of variables. The calculator shows both.
Why do I take the absolute value of the Jacobian?
Area and volume cannot be negative. A negative determinant only means the new coordinates flip orientation, so integrals use \(|\det J|\).
Can the Jacobian matrix be non-square?
Yes. Two functions of three variables give a 2×3 matrix. Only square Jacobian matrices have a determinant.
How is the Jacobian related to the gradient?
For a single function, the Jacobian matrix has one row, and that row is the gradient. The Jacobian is the gradient idea extended to several functions at once.