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Evaluate the Laplacian of the functionzx2x2 + y2+ z2p(x, y,z)-(a) directly in Cartesian coordinates, and (b) after changing to a spherical polarcoordinate system. Verify that, as they must, the two methods give the sameresult.

Question
Evaluate the Laplacian of the function
zx2
x2 + y2+ z2
p(x, y,z)-
(a) directly in Cartesian coordinates, and (b) after changing to a spherical polar
coordinate system. Verify that, as they must, the two methods give the same
result.
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Evaluate the Laplacian of the function zx2 x2 + y2+ z2 p(x, y,z)- (a) directly in Cartesian coordinates, and (b) after changing to a spherical polar coordinate system. Verify that, as they must, the two methods give the same result.

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Step 1

(a) Let us calculate the Laplacian of the function directly in Cartesian co-ordinates :

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Step 2

Now, putting the values of Equation  (ll), (lll) and (IV) in Equation (l) and on simplifying, we get :

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Step 3

(b) Let us change the function to a sph...

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