4. Multiple Integrals

Change Of Variables And Jacobians — Quiz

Test your understanding of change of variables and jacobians with 5 practice questions.

Practice Questions

Question 1

Which of the following best describes the role of the Jacobian in a change of variables for multiple integrals?

Question 2

Consider the transformation from rectangular coordinates $(x, y)$ to polar coordinates $(r, \theta)$. What is the correct expression for the Jacobian of this transformation?

Question 3

Which of the following integrals correctly represents the change of variables for the integral $\int_{0}^{1} \int_{0}^{1-x} f(x, y) \, dy \, dx$ using the transformation $u = x + y$ and $v = y$?

Question 4

In a three-dimensional transformation from $(x, y, z)$ to $(u, v, w)$, which of the following represents the volume element in the new coordinates?

Question 5

Suppose we use the transformation $x = u^2 - v^2$ and $y = 2uv$. What is the Jacobian determinant for this transformation?