C-14. In vector notation, Newton's equations for a single particle are d'r m = F(x, y, z) dt2 By operating on this equation from the left by r x and using the result of Problem C-13, show that m dr dt . X =rxF Because momentum is defined as p = mv = m-, the above expression reads dr dt dt "( rxp) = rx F But r x p =1, the angular momentum, and so we have di =rxF dt This is the form of Newton's equation for a rotating system. Notice that dl/dt = 0, or that angular momentum is conserved if r x F = 0. Can you identify r x F? Prob. C-13 is shown here C-13. Using the results of Problem C-12, prove that for information only: d'u d u x u x du d+2 dt dt C-15. Find the gradient of f (x, y, z) = x2 - yz + xz at the point (1, 1, 1)