Problems: Polar Coordinates and the Jacobian
y
∂(r, θ)
1
= .
x
2 + y 2 and θ = tan−1 . Directly calculate the Jacobian
x
∂(x, y)
r
Answer: Because we are familiar with the change of variables from rectangular to polar
∂(r, θ) ∂(x, y)
coordinates and we know that
·
= 1, this result should not come as a surprise.
∂(x, y) ∂(r, θ)
1. Let r =
∂(r, θ)
r r
=
x y θx θy
∂(x, y)
√ 2x
x2 +y 2
−y/x2
1+(y/x)2
2
=
x
x2 +y 2
− 2y 2
x +y
y
x
r
r
y
− 2 x2
r
r
x2 + y 2
√
=
=
=
r3
√ 2y
x2 +y 2
1/x
1+(y/x)2
√ y
x2 +y 2
x
x2 +y 2
2
1
= .
r
√
2.
For the change of variables x = u, y = r2 − u2 , write dx dy in terms of u and r.
∂(x, y)
Answer: We know dx dy =
du dr.
∂(u, r)
∂(x, y)
x xr
=
u
yu yr ∂(u, r)
=
=
Hence dx dy = √
r
du dr.
r2 − u2
1
√ −u
r2 −u2
r
√
2
r − u2
0
√ r
r2 −u2
MIT OpenCourseWare
http://ocw.mit.edu
18.02SC Multivariable Calculus
Fall 2010
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