28 yields 1d r dr r dr dr r d2 r r2 d2 144 we multiply

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Unformatted text preview: rdinates are related to the standard Euclidean coordinates by the formulae x = r cos θ, y = r sin θ. The tool we need to express the Laplace operator ∆= ∂2 ∂2 +2 ∂x2 ∂y in terms of polar coordinates is just the chain rule, which we studied earlier in the course. Although straigtforward, the calculation is somewhat lengthy. Since ∂x ∂ = (r cos θ) = cos θ, ∂r ∂r 136 ∂y ∂ = (r sin θ) = sin θ, ∂r ∂r it follows immediately from the chain rule that ∂u ∂u ∂x ∂u ∂y ∂u ∂u = + = (cos θ) + (sin θ) . ∂r ∂x ∂r ∂y ∂r ∂x ∂y Similarly, since ∂x ∂ = (r cos θ) = −r sin θ, ∂θ ∂θ ∂y ∂ = (r sin θ) = r cos θ, ∂θ ∂θ it follows that ∂u ∂u ∂x ∂u ∂y ∂u ∂u = + = (−r sin θ) + (r cos θ) . ∂θ ∂x ∂θ ∂y ∂θ ∂x ∂y We can write the results as operator equations, ∂ ∂ ∂ = (cos θ) + (sin θ) , ∂r ∂x ∂y ∂ ∂ ∂ = (−r sin θ) + (r cos θ) . ∂θ ∂x ∂y For the second derivatives, we find...
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