I'm learning line integrals. And I understand the concept of integration with respect to $ds$, but I don't know what $dx$ and $dy$ actually mean.
In my calc textbook, it just says that $dx$ is the same thing as $x^{'}(t)dt$ and likewise for dy being equal to $y^{'}(t)dt$.
But what do they mean physically. Intuitively I know that ds is the arc length, or a certain curtain in a sense. And I also know that if ds is present I just use the arc length formula, but for dx and dy I don't have to.
I looked at the other questions but I never understood that well.
Best Answer
$dx$ is infinitesimal change in the $x$-direction. $dy$ is an infinitesimal change in the $y$-direction. $ds$ is an infinitesimal change in arc length. Think of them in a triangle. $dx$ and $dy$ are legs of the triangle, and $ds$ is the length of the hypotenuse.
You have to use the arc length formula for calculating arc length but not change in $x$ for the same reason that you need the Pythagorean theorem to compute the length of the hypotenuse of a triangle, but not the length of a leg.
It may be helpful to realize that while $ds$ depends on the particular shape of your curve, $dx$ and $dy$ always follow the coordinate axes. However in the case that your curve is a horizontal line, (or even just has $dy/dx=0$ for a point) then $ds=dx$. And if it is a vertical line (or even just has $dx/dy=0$ for a point), then $ds=dy.$
And in the case that $ds=dx$, then $\int f(x,y)\,ds=\int f(x,y)\,dx$ is just the standard integral of one variable from single variable calculus.