Just took calc 3 "semifinal" And I have a few questions regarding concepts, which was sparked by a conversation with a few classmates after the test. 1st, a problem was given to us and we were asked if stokes or divergence applies. It was a line integral, so divergence did not apply. It said that C is a "closed loop". Everything else was textbook stokes. So I wrote that stokes applied. However, one of my classmates told me that a "closed loop" may cross its own boundary and thus doesn't enclose a single surface so stokes does not apply. I honestly did not think of that. What do you think? I think "closed loop" implies a loop, a single loop, that doesn't cross its self but I just go by how I define the word "loop." I have no mathematical reason. The last part of our test was nested things like "grad(div F)" and we were simply to write if these were a vector, scalar, or neither (nonsensical.) At the end, a question was asked "which of these is always zero?" I only had two of these nested functions as scalars, div(grad f) and div(curl F). Doing a bit of thinking, I figured the former came down to second derivatives, which isn't necessarily 0, so I picked div(curl F). However, an engineering major told me that it was definitely grad(grad F) because the first grad gives you a perpendicular vector, so a vector perpendicular to that is parallel to the original (???). I don't understand his logic at all. Gradient is del(scalar field) which is a vector field itself. Thus I put neither for grad(grad f) and didn't even consider it, because by our definitions the gradient of a vector doesn't make sense. That's not tackling his reasoning about the parallel => 0 thing..