What is an integral with a circle through it?

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The symbol ∮ represents a line integral taken over a closed curve, which is homeomorphic to a circle. It is commonly used in vector calculus to evaluate integrals around closed paths. Additionally, this symbol can extend to integrals over closed manifolds in broader mathematical contexts. Understanding its application is essential for analyzing physical phenomena, such as in electromagnetism. The integral with a circle signifies the importance of path independence in certain integrals.
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\oint what's the difference?
 
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This is a symbol usually used to denote a line integral over a closed curve (a curve homeomorphic to the circle, hence the circle symbol), but it can also be used for an integral over a closed manifold in more general circumstances.
 

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