AcademySeveral-Variable Calculus
Academy
Coordinate Changes
Level 1 - Calculus topic page in Several-Variable Calculus.
Principle
Coordinate changes rewrite an integral using variables better suited to the region. The area or volume element changes by a scale factor called a Jacobian.
Notation
\(u,v\)
new variables
\(J\)
Jacobian determinant
\(r,\theta\)
polar coordinates
The Core Method
When changing variables, transform both the integrand and the area element.
Change of variables
\[dA=|J|\,du\,dv\]
For polar coordinates,
Polar area element
\[x=r\cos\theta,\quad y=r\sin\theta,\quad dA=r\,dr\,d\theta\]
Worked Cases
Question
Use polar coordinates to find the area of a disk of radius \(a\).
Answer
The disk is \(0\le r\le a\), \(0\le\theta\le2\pi\). Area is \(\int_0^{2\pi}\int_0^a r\,dr\,d\theta\). The inner integral is \(a^2/2\). Multiplying by \(2\pi\) gives \(\pi a^2\).
Examples
Question
Why is \(dA=r\,dr\,d\theta\) in polar coordinates?
Answer
A small polar sector has radial thickness \(dr\) and arc length approximately \(r\,d\theta\). Multiplying these gives area approximately \(r\,dr\,d\theta\), so the Jacobian factor is \(r\).