Mathematics · University
Multiple integrals and Jacobians · revision notes
Draw or describe the region before changing integration order.
Key ideas and conditions
Bounds describe a region
Draw or describe the region before changing integration order.
Area scaling
Under a smooth one-to-one change of variables, the area factor is the absolute Jacobian determinant.
Polar coordinates
x=r cos θ, y=r sin θ and dA=r dr dθ for r≥0.
Orientation versus area
A signed Jacobian tracks orientation; an area integral uses its absolute value.
Explain without looking
- Explain why ignoring the Jacobian in x=2u,y=3v gives a wrong area.
- A coordinate change has determinant −5 at a point. Which factor belongs in an ordinary area integral there?
Further reading and review references
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