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Our textbook sometimes uses notation like this:

$$ \oint_C x \mathrm{d}y + y \mathrm{d}z + z\mathrm{d}x $$

and I have no idea what it means. Both the textbook and my professor's explanations make no sense to me. I understand that it is not the same as $\oint_C x \mathrm{d}y + \oint_C y \mathrm{d}z + \oint_C z\mathrm{d}x$. What does it mean?

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    $\begingroup$ Looks like a line integral. $\endgroup$ – Karl Nov 15 '17 at 20:33
  • $\begingroup$ @Karl it is indeed, I edited it to avoid confusion. $\endgroup$ – Bluefire Nov 15 '17 at 20:38
  • $\begingroup$ I'm not sure we use the same definition of "understand". Why do you think it's not the same as $\oint_C x \mathrm{d}y + \oint_C y \mathrm{d}z + \oint_C z\mathrm{d}x$? All integrals I ever met were linear, and that certainly includes additive. $\endgroup$ – Professor Vector Nov 15 '17 at 21:10
  • $\begingroup$ We might need some more context to explain this to you: are you studying differential forms? $\endgroup$ – amd Nov 15 '17 at 21:18
  • $\begingroup$ It makes sense whenever you use $()$'s. Namely, $\displaystyle\oint_{C}\left(x\,\mathrm{d}y + y\,\mathrm{d}z + z\,\mathrm{d}x\right) = \oint_{C}x\,\mathrm{d}y + \oint_{C}y\,\mathrm{d}z + \oint_{C}z\,\mathrm{d}x$. $\endgroup$ – Felix Marin Nov 15 '17 at 21:31
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  • Much simpler, let $\vec{F} = \langle F^1, F^2, F^3 \rangle$ and $\vec{r} = \langle x ,y ,z \rangle$. Then $d \vec{r} = \langle dx, dy,dx \rangle$ (recall $df = f'(t) dt$, just so this notation isn't stange). Hence, $\vec{F} \cdot d \vec{r} = F^1 dx + F^2 dy + F^3 dz$ which you can interpret as the infinitesimal work.

  • Let $\gamma(a) = P$ and $\gamma(b) = Q$. Now recall one more definintion, given $C$ is a curve and $\gamma: [a,b] \to \mathbb{R}^3$ parametrizes $C$ i.e $\gamma(t) \in C$ and $C = \textrm{Im} (\gamma)$ then,

$$ \int_C \vec{F} \cdot d \vec{r} = \int_{P}^{Q} F^1 dx + F^2 dy + F^3 dz $$

  • If $P=Q$ then $C$ is a closed curve and we use embed this notation into the integral by using the symbol $\oint$.

$$ \oint_C \vec{F} \cdot d \vec{r} = \int_{P}^{Q = P} \vec{F} \cdot d\vec{r} = \int_{P}^{Q = P}F^1 dx + F^2 dy + F^3 dz$$

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