Question

I want to learn how to do this but I need help. Can you explain what the question is asking for and hint how I should handle it? (picture)

Answer 1

Do you know how to calculate a line integral?

Answer 2

\[
\int_C \mathbf A\;ds = \int_{t_0}^{t_1}\mathbf A(\mathbf r(t))\cdot d\mathbf r = \int_{t_0}^{t_1}\mathbf A(\mathbf r(t))\cdot \mathbf r'(t)\;dt
\]

Answer 3

So I want to integrate A, why does r come into this though?

Answer 4

Okay so \(C\) is the curve you're integrating a long. \(\mathbf r(t)\) is the parametrization of that curve.

Answer 5

The parametrization allows you to change it to an integration along \(t\) which is something we can feasibly do. Actually it is better to simply think of it as a definition.

Answer 6

Ah I see! Alright let me see what I get.

Answer 7

So did you use the chain rule? Can you show what you did?

Answer 8

What do you mean? I didn't do any work.

Answer 9

Hmm I think I see what you did

Answer 10

I should have put \(d\mathbf s\) up there.

Answer 11

To which part? Also one question: When I differentiate r do I remove the e?

Answer 12

What is e? is it a unit vector?

Answer 13

Yes the unit vector. It's probably a stupid question but I forgot

Answer 14

I suppose \(\mathbf e_1\) corresponds to \(x\)

Answer 15

Do you know how to compose vector functions?

Answer 16

Now that you asked, I'm not so sure anymore. And I think I need a couple hours of sleep before school so I will read your reply when I wake up. Thank you very much in advance :)