Fundamental Theorem of Calculus and Line Integrals: Does it Apply?

[LucasGB]

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.

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 of vector fields.

 

[elibj123]

A scalar field has no "anti-derivative" as it is a function of several variables, just like it doesn't have a derivative, but only a partial derivative.

Fundamental Theorem of Calculus is only about one variable functions f(x).
In integration of a curve, you choosed a parametrization and then your integrand reduces to a single-variable function g(t), and then the fundamental theorem applies.

So you should really say that the fundamental theorem of calculus is not really relevant to your question.

 

[LucasGB]

A scalar field has no "anti-derivative" as it is a function of several variables, just like it doesn't have a derivative, but only a partial derivative.

Fundamental Theorem of Calculus is only about one variable functions f(x).
In integration of a curve, you choosed a parametrization and then your integrand reduces to a single-variable function g(t), and then the fundamental theorem applies.

So you should really say that the fundamental theorem of calculus is not really relevant to your question.

Allright, that's interesting, but I still have doubts.

1. Let's say I want to do a surface integral to find the mass of a surface. So I simply integrate p (area density) with respect to dA over the entire surface. I can say p is a scalar field, since it is a value attributed to all points of the surface. Can't I say mass is the antiderivative of p, since p=dm/dA?