$\int_{-c}\mathbf{F}.\mathbf{dl}=-\int_{c}\mathbf{F}.\mathbf{dl}$-what is wrong here?

Question

We know about line integral that $\int_{-c}\mathbf{F}.\mathbf{dl}=-\int_{c}\mathbf{F}.\mathbf{dl}$. Suppose my $\mathbf{F}$ is $\frac{\mathbf{r}}{r^3}$ and path is radial path from $r=a$ to $r=b$. so $\int_{c}\mathbf{F}.\mathbf{dl} =\int_{a}^{b}\frac{\mathbf{r}}{r^3}.\mathbf{dr}= \frac{1}{a}-\frac{1}{b}.$But if i take the path directly opposite to the given path we will get $\int_{-c}\mathbf{F}.\mathbf{dl} =\int_{b}^{a}\frac{\mathbf{r}}{r^3}.\mathbf{-dr}= \frac{1}{a}-\frac{1}{b}$.What is wrong in here?Is it problem with limit?I think my limit is wrong.it will be very helpful if anybody explain conceptwise and also physically

Answer 1

When I used to teach this material, I used to tell my students to determine the sign of the answer first, conceptually, and then make sure not to let the formalism interfere with what you know to be correct.

So for example, if ${\bf F} = {\bf r}/r^3$, and if the path points with ${\bf r}$, then you know that each infinitesimal bit of work ${\bf F}\cdot d{\bf l}$ is positive. So whatever you do with the integral, do make sure that you are adding up a bunch of positive numbers! Likewise, if the path points against ${\bf r}$, make sure to set up an integral that adds up a bunch of negative numbers.

You basically have a choice of whether you are going to handle the direction of the path in the expression for $d{\bf l}$, or in the order of the limits of integration. But you can't handle it in both places at once, or your choices cancel out. That is, you can adopt the idea that ${\bf F}\cdot d{\bf l} = F_rdr$, and handle directions by ordering the limits of integration; or you can adopt the idea that $d{\bf l}$ should be $-\hat{\bf i}dr$ in the reversed path, but then don't also reverse the limits of integration; that is double-counting the direction.