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?

  • 1
    $\begingroup$ 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
  • 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 $\oint$.

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.