Quote:
Originally Posted by leo255
I'm currently taking these right now. Having a bit of a hard time, for some reason. I got a 'B' in Calculus II, which most people have told me was/is the hardest out of the calculus sequence. As far as Calc III, right now we're doing contours of 3D shapes. In L.A., we did vector dot & cross, as well as got into the Cauchy–Schwarz Inequality proof.

You might want to contrast and compare the two techniques of proving the CauchySChwartz theorem.
The one using lambda technique brings you back to the discriminant method of quadratic equations in high school. It is more fun than the other.
THE INSTRUCTOR MIGHT JUST THROW THIS QUESTION ON THE FINAL EXAM, saying : "
Prove the C S Thereom ,
10 points "
As for the dot product, it is the big gun you would use when you want to prove Orthogonality.
Conclusion: Keep in mind Calc III is just an extension of calc I. For instance the idea of stationary points in 3D generalizes the concept of smooth critical points in one dimension.The same way derivatives are zero at smooth critical points in 1D, the same way all three partial derivatives are zero at smooth stationary points in 3D ).
As far as 3D integration is concerned, FUBINI Theorem is the key, it is nothing but the generalization of the technique of " changing variable " in Calc I (when using one dim. Integration).
Pay attention to the Jacobian and the change in the boundary of the region.
Good luck.