In the context of integrals and differential equations, often the symbol $df$ or $dy$ appears, where in some previous steps $f$ and $y$ were functions. What do these symbols mean $df$ and $dy$?
Some examples where $dx$ appears is:
the first fundamental form of a surface, see first fundamental form . Or By computing the winding number of a curve defined by $c:J \to \mathbb{R^2}, t \mapsto(x(t),y(t))$. Then $$\frac{1}{2\pi} \oint_c \,\frac{x}{r^2}\,dy - \frac{y}{r^2}\,dx $$,where $r^2=x^2+y^2$.
Does this mean $$\frac{1}{2\pi} \oint_c \,\frac{x}{r^2}\,dy - \frac{y}{r^2}\,dx=\frac{1}{2 \pi}\int_J \frac{x(t)}{x(t)^2+y(t)^2}\cdot y'(t) dt -\frac{1}{2\pi} \int_J \frac{y(t)}{x(t)^2+y(t)^2}\cdot x'(t) dt$$?
Thanks for the help
bests
bjn
It means a small change in the value of function $f$. You slice up your function curve in infinite possible elements, and then each of will be $df$ in length, and integrating w.r.t. $df$ or $dy$ then means you are summing up all those small slices. This way we can find area under a curve or length of a curve using double or single integrals, also volume using triple integrals.