(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

"The flow of a fluid past a wedge is described by the potential

ψ(r,θ) = -crsin^{α}(αθ),

wherecandαare constants, and(r,θ)are the cylindrical coordinates of a point in the fluid (the potential is independent ofz). Verify that this function satisfies Laplace's equation,Δψ = 0."

2. Relevant equations

Δ is the Laplacian operator

3. The attempt at a solution

I have proved the statement, but I feel like I had to fudge the Chain Rule in order for it to work out. I have that δ/δr(ψ) =-αcrsin^{α}αθbut I think that since α is some constant I ought to have δ/δr(ψ) =-αcrsin^{(α-1)}αθ, and the second partial derivative would yield-α(α-1)crsin^{(α-2)}αθ. If done this way, the terms will not cancel and I cannot conclude the Laplacian is equal to zero.

Why doesn't the power of α change if it's a constant? I thought that d/dx(x^{n}) = nx^{(n-1)}, so why does that not seem to apply in this case?

Thanks for any insight.

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# Chain-rule issue on Laplacian equation

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