Let be an open set in and be a continuously differentiable scalar field. Let

and let , be any smooth curve in such that initial point of is and final point of is . Then

Let be a continuously differentiable vector-field such that is conservative, i.e.,

for some continuously differentiable scalar field on . Then, for and for any smooth curve

Consider the function

Then is a continuously differentiable function with

Hence, by fundamental theorem of calculus for a single variable,

(ii) This follows from (i).

We give next some applications of this theorem.

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