If I draw a random curve over a scalar field, then it is not generally true that the line integral of the scalar field over the curve equals the difference between the value of the antiderivatives of the scalar field at the beginning and finishing points of the curve, as one can clearly see by changing the shape of the curve and keeping the aforementioned points unchanged.(adsbygoogle = window.adsbygoogle || []).push({});

If, on the other hand, the scalar field is a linear mass density and the curve is a piece of wire, then it is true that the line integral of the scalar field over the curve equals the difference between the value of the antiderivatives of the scalar field at the beginning and finishing points of the curve, because now the scalar field is closely related to the curve, and if I change the shape of the curve, I'll change the scalar field in the same manner. (Remember that the antiderivative of a linear mass density is mass)

So, should I say the Fundamental Theorem of Calculus applies or doesn't apply to line integrals?

PS.: Note I'm not talking about the Fundamental Theorem of Line Integrals, which is all about the line integrals ofvectorfields.

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# Integral puzzle

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