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Multivariable Calculus- Can someone help me ? On the problem attached in the image, I cannot get the same answer : zero.

Which topic should I look up on solving this issue?

Please help.

https://nimb.ws/f3Gdhz

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1 Answer 1

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The topic is "line integral": https://en.wikipedia.org/wiki/Line_integral

It is also important to understand (eventually) the physics behind it: http://www.feynmanlectures.caltech.edu/II_03.html

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  • $\begingroup$ Thank you. The Caltech article, which I could not find in the past, is good and well written. On the other hand, I wonder what your answer will be and if you come up with the same answer ,"zero" as shown in the solution in the image. $\endgroup$
    – Big Smile
    Mar 18, 2019 at 22:49
  • $\begingroup$ I am getting a zero as well. Here is how: parameterize the semi-circle by: $x(t) = \cos(t), y(t)=0, z(t) = \sin(t)$, $\pi < t < 2\pi$. Then $$ F = (0, \cos^3(t), \cos^2(t)), $$ and $$ dr = (-\sin(t), 0, \cos(t)) dt, $$ so $$ F \cdot dr = \cos^3(t), $$ which (noting that $t = 3\pi/4$ is the midpoint of our integration interval) has the property of being odd w.r.t. that midpoint: $$ \cos^3(t - 3 \pi/4) = - \cos^3(-t - 3 \pi/4), $$ so the integral is zero. $\endgroup$
    – avs
    Mar 19, 2019 at 22:17
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    $\begingroup$ Great explanations! Thanks a lot! $\endgroup$
    – Big Smile
    Mar 21, 2019 at 9:31
  • $\begingroup$ $3 \pi/2$, not $3\pi /4$. $\endgroup$
    – avs
    Mar 21, 2019 at 16:44

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