Calculus Volume 3 · free preview
Change of Variables in Multiple Integrals
The change-of-variables theorem replaces the integrand and the element by the transformed versions, the absolute Jacobian determinant correcting the area or volume, unifying polar, cylindrical, and spherical as special cases.
1. The two-variable theorem
The two-variable theorem
2. Computing Jacobians
Computing Jacobians
3. Three variables and the theorem's scope
Three variables and the theorem's scope
4. Choosing the transformation
Choosing the transformation
5. Change of variables formula
The double integral's change of variables theorem introduces which factor?
6. Polar Jacobian computation
Computing the Jacobian of x = r cos theta, y = r sin theta directly gives which value?
7. Inverse Jacobian
The Jacobian of the inverse transformation is which expression?
8. Three-variable Jacobian
The spherical coordinates' three-variable Jacobian equals which expression?
9. Why absolute value
The change-of-variables formula uses the absolute value of the Jacobian rather than the signed value because of what?
10. Jacobian sign handling
The change-of-variables formula uses the absolute value of the Jacobian. As a sign, what is taken? Answer with one word.
11. Jacobian determinant size two variable
The two-variable Jacobian is a determinant of size two by which number? Answer with a number.
12. Transformation regions
In the change of variables, the original region R corresponds to S in the new coordinates. As a single letter, what is S? Answer with an uppercase letter.
13. Linear map uniform scaling
For a linear change of variables, the Jacobian is constant, and the integral over a region scales by that constant times what?
14. Linear map scaling
A linear transformation scales all areas by the absolute Jacobian's constant. As a single word, what is it? Answer in lowercase.
Practice this interactively
Free account · instant grading · spaced review that schedules itself.
Start this course for free