What does an integral symbol with a circle mean?

Question

I have frequently seen this symbol used in advanced books in physics:

$$\oint$$

What does the circle over the integral symbol mean? What kind of integral does it denote?

Answer 1

It's an integral over a closed line (e.g. a circle), see line integral.

In particular, it is used in complex analysis for contour integrals (i.e closed lines on a complex plane), see e.g. example pointed out by Lubos.

Also, it is used in real space, e.g. in electromagnetism, in Faraday's law of induction (part of the Maxwell equations, written in an integral form):

$$\oint_{\partial \Sigma} \mathbf{E} \cdot d\boldsymbol{\ell} = - \int_{\Sigma} \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A} $$ saying that the generated voltage (an integral of electric field along a circle) is the same as the time derivative of the magnetic flux.

Answer 2

It's an integral over a closed contour (which is topologically a circle). An example from Wikipedia: $$ \begin{align} \oint_C {1 \over z}\,dz & {} = \int_0^{2\pi} {1 \over e^{it}} \, ie^{it}\,dt = i\int_0^{2\pi} 1 \,dt \\ & {} = \Big[t\Big]_0^{2\pi} i=(2\pi-0)i = 2\pi i, \end{align} . $$

Answer 3

Twelve years after this question was asked, a more comprehensive answer covering cases not mentioned previously is still needed.

The usual usage of the $ \unicode{x222e}~$ symbol in Mathematics denotes an integral over a closed line. The traditional symbols for integrals over closed surfaces or volumes$^{(*)}$ were $ \unicode{x222F}~$ and $ \unicode{x2230}$. However, different usages can be found in Physics.

In particular, in some cases, the $ \unicode{x222e}~$ symbol was used with a different meaning. For example, Landau and Lifschitz, Sommerfeld, and other authors also use it to denote the integral over a closed surface in electromagnetism.

Therefore, when reading the Physics literature, it is recommended to carefully check the domain of integration without sticking to the $ \unicode{x222e}~$ symbol alone.

$^{(*)}$: a closed volume is a closed (compact and without boundary) manifold in 3D. In such a case, a special symbol for integration is used less than in other cases.

Answer 4

It basically means you are integrating things over a loop. For e.g. a circle with an element $\text{d} \textbf{l}$ if you do $\oint{\text{d} \textbf{l}}$ it will give you circumference of the circle.

Answer 5

The ∮ sign denotes a "closed integral". Thus, it is used for the area of a closed surface (eg., when finding the flux of a vector field across the surface). It is also used to represent line integrals where the line is a loop. The main idea is that it denotes something bounded.