# ar/ad= 0. (e) Substituthree parts into the rigshow that you get the lethree positive inte gers (1, 2, 3, . ). [Hint: Use sepa-ration of variables, and seek a solution of the form= X(x) Y (y) Z (z). Note that by setting a b = cobtains the cubical box of Example 8.2.]SECTION 8.5 (The Three-DForce Problem)SECTION 8.4 (The Two-Dimensional Central-Force Problem)8.20 The spherical polarfined in Fig. 8.11. Derivefor x, y, and z in terms ofexpressions for r, 0, and8.16 (a) For the two-dimensional polar coordinatesdefined in Fig. 8.7 (Section 8.4), prove the relationsX = r cos and y r sin (8.104)8.21 A point P on the earcoordinates (x, y, z) and(r, 0, ø) (with coordinateis at the earth's center aWhat are the coordinatesof the place Q at the oppoCter through P?(b) Find corresponding expressions for r and interms of x andу.A certain point P in two dimensions has rectangularcoordinates (x, y) and polar coordinates (r, ). Whatare the polar coordinates of the point Q(-x, -y)?Illustrate your answer with a picture.8.17Changes of coordinates in two dimensions (such asthat from x, y to r, ) are much more complicatedthan in one dimension. In one dimension, if we have afunction f(x) and choose to regard x as a function ofsome other variable u, then the derivative of f withrespect to u is given by the chain rule,8.188.22.Substitute the separatecinto the Schrödinger equatyou multiply through byarrange, you get an equat(function of r and 0). Expladf

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For Problem 8.16, how do I prove the relations and give the correct expressions?

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Step 1

Draw a figure to represent a point using both polar and Cartesian coordinates.

Step 2

(a)

Consider the right angled triangle OMP.

Write an expression for cosϕ using trigonometry

Step 3

Write an expression for sinϕ u...

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