Math 234 section 302/307
We want to show that:
Notes on polar coordinate
Z Z
Z Z
f (x, y)dxdy =
f (r, θ)rdrdθ
We just need to show that dxdy = rdrdθ.
We know x = r · cosθ, y = r · sinθ, then by product rule:
dx = d(r · cosθ) = cosθ · dr − r · sinθ · dθ
dy = d(r · sinθ) = sinθ · dr + r · cosθ · dθ
dx, dy can be considered as two vectors vertical to each other, then dxdy is the area of the
rectangle R with two sides dx, dy.
But we know, by cross product, the area of a rectangle equals to |dx × dy|. (here × means
cross product)
Thus we have:
dxdy = |dx × dy|
= |(cosθ · dr − r · sinθ · dθ) × (sinθ · dr + r · cosθ · dθ)|
= |cosθ · sinθ · dr × dr − r · sinθ · sinθ · dθ × dr+
cosθ · r · cosθdr × dθ − r · sinθ · rcosθ · dθ × dθ|
because the cross product with itself equals to 0, then dr × dr = dθ × dθ = 0. Then:
= | − r · sinθ · sinθ · dθ × dr + cosθ · r · cosθdr × dθ|
and also we have dr × dθ = −dθ × dr.T hen :
= |r(cos2 θ + sin2 θ)dr × dθ|
= rdrdθ
This proves the polar coordinate transformation.