For x = ρsinφcosθ; y = ρsinφsinθ; z = ρcosφ: a/ Express ∂^2/∂y^2 with ρ, φ,...

For x = ρsinφcosθ; y = ρsinφsinθ; z = ρcosφ:

a/ Express ∂^2/∂y^2 with ρ, φ, θ, and their partials.

b/ Express ∂^2/∂z^2 with ρ, φ, θ, and their partials.

c/ Express the Laplacian operator using spherical coordinates.

Expert Solution

milcah answered 2 years ago

(a) Apply the chain rule to express ∂/∂ρ, ∂/∂ϕ, and ∂/∂θ using ∂/∂x, ∂/∂y, and ∂/∂z....

(a) Apply the chain rule to express ∂/∂ρ, ∂/∂ϕ, and ∂/∂θ using ∂/∂x, ∂/∂y, and ∂/∂z. (b) Solve algebraically for ∂/∂x, ∂/∂y, and ∂/∂z with ∂/∂ρ, ∂/∂ϕ, and ∂/∂θ when ρ does NOT equal 0 and sin ϕ does NOT equal 0. (Hint: you can use method of elimination to reduce the number of variables.) (c) Express ∂2/∂x2 with ρ, ϕ, θ, and their partials.

Given a function φ(z) with z = x+iy let U(x, y) = ½ [φ(x+iy) +...

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Question

For x = ρsinφcosθ; y = ρsinφsinθ; z = ρcosφ: a/ Express ∂^2/∂y^2 with ρ, φ,...

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