Consider the spherical coordinate transformation P_3(rho, phi, theta) = (rho cos

Question

Consider the spherical coordinate transformation P_3(rho, phi, theta) = (rho cos theta sin phi, rho sin theta sin phi, rho cos phi). Calculate P^*_3(dx Lambda dy Lambda dz). (It would save you a few calculations to use Theorem 3.14, but it might be more interesting for you to just calculate it directly.) 4. Let g(x) be a function of one variable, but view g as a map from R to R (with x being the coordinate in the domain and y being the coordinate in the codomain). In this case, Theorem 3.11 tells us that, for any function f(y), g^*(df) = d(g^* f). Expand both sides of this equation to obtain the chain rule formula from Calc I.

Explanation / Answer

given

P3=(cossin+sinsin+cos)

and P3*(dxdydz) = (cossin+sinsin+cos)*(dxdydz)

= cossin*(dxdydz) +sinsin*(dxdydz)+cos*(dxdydz)

= (cossindx)*(cossindy)* (cossindz) +(sinsindx)*(sinsindy)* (sinsindz)+(cosdx)*(cosdy)* (cosdz)

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