**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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**Solution:**

(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

Similarly

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

Therefore,

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