Question

Describe how to use the Jacobian to change variables in double integrals.

Short answer

The Jacobian determinant allows transformation of variables in an integral. It is applied by setting up the integral in new variables, defining and calculating the Jacobian matrix and its determinant, and applying it within the integral. The new limits of integration are used to calculate the integral, thus effectively using the Jacobian to change variables in double integrals.

Step-by-step solution

Define Variables

Let's assume a transformation from Cartesian coordinates \((x, y)\) to polar coordinates \((r, θ)\). In this case, the transformation will be \(x = r \cos θ \) and \(y = r \sin θ\). These are the new variables.

Define Jacobian Matrix

The Jacobian matrix \(J\) of this transformation would be the matrix of all first order partial derivatives. So here, \(J\) would be given by \(\[ \frac{\partial(x, y)}{\partial(r, θ)} \] = \[\[ \frac{\partial x}{\partial r}, \frac{\partial x}{\partial θ} \], [\frac{\partial y}{\partial r}, \frac{\partial y}{\partial θ} \] \]\). So, plug in our transformation equations into this expression to get the actual Jacobian matrix.

Compute the Determinant

The determinant of the Jacobian matrix, noted as \(|J|\) or \(\text{det} J\), is what will be needed in the integrals. Here, \(\text{det} J = \frac{\partial x}{\partial r} \frac{\partial y}{\partial θ} - \frac{\partial x}{\partial θ} \frac{\partial y}{\partial r}=\) once substituting our transformations, we get \(r\). This is the Jacobian determinant, often seen in polar integral setups.

Apply the Jacobian determinant in Integral

Finally, when we set up our double integral in terms of \(r\) and \(θ\), we will introduce the Jacobian determinant. We get: \(\int_a^b \int_c^d f(x, y) \, dx \, dy = \int_a^b \int_c^d f(r \cos θ , r \sin θ) \cdot |J| \, dr \, dθ\) where |J| is the determinant of the Jacobian matrix that we found earlier.

Evaluate the Integral

Evaluate the integral with the new limits of integration.