Verify that the function u=1-1/y sin(x²) cos(y²) sin(z²) is a solution of the three-dimensional Laplace equation Uₓₓ+Uᵧᵧ+U =0. (a) Define and explain the Laplace equation in three dimensions and its significance in partial differential equations.
(b) Discuss the components of the given function u and how they contribute to the Laplace equation.
(c) Outline the steps and calculations involved in verifying that u satisfies the Laplace equation.