What situations should $\oint$ be used?

Question

Sometimes people put a circle through the integral symbol: $\oint$

What does this mean, and when should we use this integration symbol?

Answer 1

This symbol is used to indicate a line integral along a closed loop. if the loop is the boundary of a compact region $\Omega$ we use also the symbol $ \int_{\delta \Omega} $

we can generalize such notation to the boundary of a region in an n-dimensional space and, if $\Omega$ is an orientable manifold we have the generalized Stokes' theorem $$ \int_{\delta \Omega}\omega=\int_ \Omega d\omega $$ that is a beautiful generalization of the fundamental theorem of calculus.