How to find the Jacobian matrix and determinant for two variables

When we switch from one pair of coordinates to another, a small region can change shape and size. The Jacobian matrix tells us how the coordinates change nearby, and its determinant gives us the local area scale.

Put the four partial derivatives in a matrix

Suppose $x = x (u, v)$ and $y = y (u, v)$ . Put the derivatives of $x$ in the first row and the derivatives of $y$ in the second row. The columns are for $u$ and $v$ , in that order.

D = [\begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{matrix}]

The subscript on a derivative tells us which input varies. For example, $x_{u}$ means differentiate $x$ with respect to $u$ while holding $v$ fixed.

Find the determinant for a two-variable transformation

Let $x = u v$ and $y = 2 u - v^{2}$ . Find the two partial derivatives of each output and put them into the matrix.

\begin{array}{l} x_{u} = v, x_{v} = u \\ y_{u} = 2, y_{v} = - 2 v \end{array}

D = [\begin{matrix} v & u \\ 2 & - 2 v \end{matrix}]

For this two-by-two determinant, multiply along each diagonal and subtract the second product from the first.

\begin{array}{l} J = x_{u} y_{v} - x_{v} y_{u} \\ = v (- 2 v) - u (2) \\ = - 2 v^{2} - 2 u \end{array}

Use the absolute value for area

At $(u, v) = (1, 1)$ , the determinant is $- 4$ . The minus sign tells us the transformation reverses orientation there. The absolute value $| - 4 | = 4$ tells us that a very small region near that point maps to about four times its area.

Near (1, 1), a small region maps to about four times its original area.

For a change of variables in a double integral on a region where the transformation is one-to-one and has a nonzero Jacobian in the interior, use $d A = | J | d u d v$ .

The same arrangement of rows and columns works with three outputs. See how to find the Jacobian in three variables.

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How to find the Jacobian matrix and determinant for two variables

Put the four partial derivatives in a matrix

Find the determinant for a two-variable transformation

Use the absolute value for area

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