Problem 1B-06**: Partial differentiation involving two-dimensional polar coordinates
(a) A function F is expressed in two dimensional polar coordinates as F(r, θ). Calculate the partial differentials in rectangular coordinates, ∂F/∂x, and ∂F/∂y.
(b) A function G is expressed in two dimensional rectangular coordinates as G(x, y). Calculate the partial differentials in polar coordinates, ∂G/∂r and ∂G/∂θ.
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Solution:
(a) From Problem 1B-4, we know
Using Eq.(1) and Eq.(1A), we have
Since
Similarly
Substituting Eqs.(3), (5) and (6) into Eq.(2), we have
(b)
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