Mathematics · University
Multiple integrals and Jacobians
Draw or describe the region before changing integration order.
Prerequisites
Single-variable integration and determinants.
Study notes
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.
Worked examples
Integrate xy over 0≤x≤2 and 0≤y≤3.
The product of integrals is (2²/2)(3²/2)=9.
9
For x=2u and y=3v, find the absolute Jacobian determinant.
The derivative matrix is diag(2,3); its determinant is 6.
6
Check your understanding
What area factor accompanies polar coordinates?
- 1/r
- 1
- r²
- r
Find the area of a disk of radius 3. Enter the coefficient of π.
Reverse the order in the integral with 0≤x≤1 and 0≤y≤x, keeping the integrand arbitrary.
Integrate x+y over the unit square.
Answers and marking guidance
Answers and marking notes
- r
The magnitude of the planar polar Jacobian is r.
- 9
Integrating r dr dθ gives πR²=9π.
- The same triangular region is 0≤y≤1 and y≤x≤1, so integrate x from y to 1, then y from 0 to 1.
Credit correct limits describing the same triangle, not a rectangle.
- 1
The x part contributes 1/2 and the y part contributes 1/2.
Further reading and review references
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