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Let $f:R^3->R$ be a continuous, and let $C_{1},C_{2}$ be two smooth curves joining two points $p,q$. Then prove that line integrals of $f$ with respect to $C_{1}$ and $C_{2}$ are equal.

My progress: To be honest, I am quite unsure whether this statement is true or not. I have tried to find counter examples, but in each time, I was finding equality for each two points with different curves. I would be glad if anyone could help me to solve this problem.

BlizzardWalker
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1 Answers1

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Take $f(x,y,z)=1$ and let $C_1$ and $C_2$ be two different curves of different lengths. Then the line integral of $f$ along $C_1$ equals the length of the curve $C_1$. Similarly, the line integral of $f$ along $C_2$ equals the length of the curve $C_2$. Thus, these integrals must be different.

Pawel
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