1. Taylor prob. 4.22 4.22 The proof in Example 4.5 (page 119) that the Coulomb force is conservative is considerably simplified if we evaluate V x F using spherical polar coordinates. Unfortunately, the expression for VF in spherical polar coordinates is quite messy and hard to derive. However, the answer is given inside the back cover, and the proof can be found in any book on vector calculus or mathematical methods.15 Taking the expression inside the back cover on faith, prove that the Coulomb force F = y/r2 is conservative. Section 4.4 The Second Condition that F be Conservative It can be shown (though I shall not do so here³) that a force F has the desired property, that the work it does is independent of path, if and only if V x F = 0 (4.36) everywhere. The quantity V x F is called the curl of F, or just "curl F," or "del cross F." It is defined by taking the cross product of V and F just as if the components of V, namely (a/ax, a/ǝy, a/əz), were ordinary numbers. To see what this means, consider first the cross product of two ordinary vectors A and B. In the table below, I have listed the components of A, B, and A x B: vector x component y component Z component Ax Bx Ay By Az (4.37) B₂ A B A x B A,B₂- AzBy A₂Bx - AxB₂ AxBy - A,Bx The components of V x F are found in exactly the same way, except that the entries in the first row are differential operators. Thus, vector x component y component Z component ▼ F д/дх Fx VXF FF д/ду Fy FF a/az (4.38) F₂ F-Fx No one would claim that (4.36) is obviously equivalent to the condition that F.dr is path-independent, but it is, and it provides an easily applied test for the path-independence property, as the following example shows. 119