Practice question · Sort into groups
Sort each statement as True or False about the Jacobian.
Groups: True · False
- For Cartesian-to-Cartesian, (identity)
- The Jacobian for polar coords is
- The Jacobian is a determinant of partial derivatives
J scales area/volume elements under coordinate change - A Jacobian of 0 means the transformation collapses dimension
- The Jacobian is always positive
Hints
- The determinant can be negative, but an integral cannot use it raw.
- Orientation reversal is why the absolute value is taken.
Show the answer
True: The Jacobian is a determinant of partial derivatives,
| J | scales area/volume elements under coordinate change |
False: The Jacobian for polar coords is , The Jacobian is always positive
Why
The Jacobian can be negative (orientation-reversing); we take |J| for integrals.
Practise Change of Variables and the Jacobian
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