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What's the difference between $\int_C f\,ds$ and $\int_C F \cdot dr$?

And is $\int_C P \, dx+\int_C Q\,dy$ just notation for $\int_C f\,ds$?.

I am referring here to Section 13.2 from James Stewart's Essential Calculus and trying to understand this very basic thing. It seems like there is this notion of dot product when thinking about integrating along a path in a vector field, and then this notion of integrating with respect to arclength.

Is that correct?

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In the first integral you integrate a real valued function; in the second you integrate a vector valued function. I do not consider complex integration for simplicity.

Let $C:[a,b]\rightarrow\mathbb R^2, s\mapsto C(s):=(C_1(s),C_2(s))$ be a smooth curve in $\mathbb R^2$, and $f:\mathbb R^2\rightarrow\mathbb R$ a real valued function.

Then the integral of $f$ along $C$ is denoted by $\int_C f$ or $\int_C f ds$ (this second notation is a bit spurious) and it is equal to

$$\int_C f:=\int_a^b f(C(s))\|\frac{dC}{ds}\|ds, $$

where $$\|\frac{dC}{ds}\|^2:=\left(\frac{dC_1}{ds}\right)^2+\left(\frac{dC_1}{ds}\right)^2.$$

If $F:\mathbb R^2\rightarrow \mathbb R^2$ is a vector valued function, then the integral denoted by $\int_C F\cdot dC$ is equal to

$$\int_C F\cdot dC:=\int_a^b F(C(s))\cdot\frac{dC}{ds}ds, $$

where $\cdot$ is the scalar product of vectors in $\mathbb R^2$, and it is often called the work of $F$ along $C$. You can generalize the work integral in $\mathbb R^3$, of course. Heuristically, you are considering the equality

$$dC"="\frac{dC}{ds}ds$$

in the definition of $\int_C F\cdot dC$, where $dC$ denotes an infinitesimal displacement along $C$.

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