Problem1B-07***: Partial differentiation related to three-dimensional polar coordinates
(a) A function F is expressed in three dimensional polar coordinates (r, θ, φ). Calculate the partial differentials ∂F/∂x, ∂F/∂y and ∂F/∂z in rectangular coordinates .
(b) A function G is expressed in three dimensional rectangular coordinates (x, y, z). Calculate the partial differentials ∂G/∂r, ∂G/∂θ and ∂G/∂φ.
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(a) From Eqs.(1) and (2) of Problem1B-5, we know
From calculus, we have
Partial differentiating Eqs.(1A) and then applying Eqs.(1), we get
To calculate ∂θ/∂x, first use Eqs.(1A) and then Eqs.(1) to get
Performing similar steps for ∂θ/∂y, we get
For ∂θ/∂z, we get
Substituting Eqs.(3), (4A), (4B), (4C), (5A), (5B), and (5C) into Eq.(2), we get
(b) We know from calculus that
From Eqs.(1) we get
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