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Math Help - Line Integrals

  1. #1
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    Line Integrals

    Hi guys, I was reading over the fundamental theorem of line integrals and I have a question.

    The theorem says "Let F = Mi + Nj + Pk be a vector field whose componenents are continuous throughout an open connected region D in space. Then, there exists a differentiable function f such that F= \nabla f iff for all points in D, the values of the integral from A to B of F dot dr is independent of the path from A to B; if this is so then its value is f(B)-f(A)."

    In the proof of this, it says "Let C be a smooth curve in D that connects points A and B be defined by r(t)=g(t)i + h(t)j + k(t)k, a \leq t \leq b. Along the curve, f is a differentiable function of t."

    It then goes on to use chain rule, definition of dot product, fundamental theorem of calculus, etc. My question is...how do we know that along the curve C, f is a differentiable function of t?!

    Any insight would be much appreciated.
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