Section 12-6

course Mth 275

Section 12.6 I found this section interesting. It makes sense as an extension of the interpretations of derivatives of a single variable. I had a problem with only one of the questions, #57: which gives a formula for Newton’s law of universal gravitation and says to show that d/dx (1/r) = -x/r^3, and similarly for d/dy and d/dz, given that r^2=x^2+y^2+z^2. I was unable to get anywhere close to this using partial differentiation. I had no problem with the proofs in the other problems.

1 / r = (x^2 + y^2 + z^2)^-(1/2) so the derivative with respect to x is -1/2 * 2x / (( x^2 + y^2 + z^2) ^(3/2)) = - x / (sqrt(x^2 + y^2 + z^2)^3) = - x / r^3.

Very similar steps give the results for the y and z derivatives.