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Laplacian of a function under different coordinates

Please assist with the attached problem.
(a) Calculate the Laplacian of function u(x,y,z) = x^3 - 3xy^2 + z^2 in 3D Cartesian coordinates.
(b) Convert the formula for u into formula for u involving cylindrical polar coordinates. Then compute the Laplacian using the cylindrical polar form. Show that your answer here is the same as your answer from (a).
(c) For this section take u(x,y,z) =x ^2, Convert the formula for u into a formula for u involving spherical polar coordinates. Then compute the Laplacian using the spherical polar form. Check that your answer is 2.

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Solution Summary

The solution shows detailed calculation of Laplacian in three different 3D coordinate systems: Cartesian (rectangular), cylindrical polar, and spherical.

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