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Change to polar coordinates

Partial derivatives & multiple integrals

Japanese school year: University year 1

What you learn

Learn how to transform double integrals from Cartesian to polar coordinates using the Jacobian determinant. This technique is particularly useful for simplifying integrals over circular or symmetric regions. Before studying this topic, familiarity with basic double integrals and trigonometry is recommended.

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Key points

These equations convert Cartesian coordinates (x,y)(x, y) into polar coordinates: distance rr from the origin and angle θ\theta. Use them to simplify problems with circular symmetry.

x=rcos⁡θ,y=rsin⁡θx = r\cos\theta, \quad y = r\sin\theta

When changing variables to polar coordinates in an integral, the area element dxdydxdy transforms into r dr dθr\,dr\,d\theta. Remember to include the extra factor rr.

dxdy=r dr dθdxdy = r\,dr\,d\theta

This formula evaluates double integrals over circular regions by transforming into polar coordinates. Substitute xx and yy, and integrate with the extra factor rr.

∬Df(x,y) dxdy=∬D∗f(rcos⁡θ,rsin⁡θ) r dr dθ\iint_D f(x,y)\,dxdy = \iint_{D^*} f(r\cos\theta, r\sin\theta)\,r\,dr\,d\theta

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