Practice question · Multiple choice
When you switch a double integral to polar coordinates, becomes , not just . What does that extra pay for?
Hints
- Draw two polar grid cells with the same : one near the origin, one far away. Compare their sizes.
- A cell spans radially and tangentially, its area is the product.
Show the answer
D. Area cells far from the origin are bigger
Why
The polar grid’s cells grow with distance: width times depth . The factor , the Jacobian of the transformation, is the local area-conversion rate. Forgetting it is the most common change-of-variables error in the subject.
Practise Change of Variables and the Jacobian
The app has 8 more questions on this lesson, and keeps your place in the course. Physics I is free to start.
More questions on Change of Variables and the Jacobian
- Sort each statement as True or False about the Jacobian.
- The Jacobian appears as a multiplying factor in every change of variables. What would go wrong if it were…
- Order the steps of a change of variables in a double integral.
- Match each coordinate change to its volume/area factor (its Jacobian).
- Using polar coordinates, ∬ r dr dθ over the full disk of radius 2 equals 2π × ∫ 0² r dr. Evaluate the area of…