Understanding Stokes' Theorem and Gradient Functions

[1MileCrash]

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 itself 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..

 

[tiny-tim]

hi 1MileCrash!

… 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 itself but I just go by how I define the word "loop."

if your classmate defines a loop as the image of a function from a circle onto the space, then yes he's right …

but then "loop" would also have to include a completely squashed loop (every point repeated, except for two "ends"), and anything in between …

so i think your definition is the more sensible one ​

… "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 … didn't even consider it, because by our definitions the gradient of a vector doesn't make sense.

yes, no such thing as grad of a vector