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Directed Surfaces and Curves

Take surface S bounded by closed curve $\Gamma$. At every point on S there are two possible normal vectors.

 

 

Assign direction to $\Gamma$ and use the right-hand rule.

A closed surface has no boundary. Actually it has no unique boundary.

 

It means that for any chosen boundary $\Gamma$ one should take opposite directions for the two parts of the closed surface bounded by $\Gamma$.

 

Divergence, Gradient and Curl

 

Next, Fundamental Theorems of Vector Calculus