Can some one explain to me what is a curve integral of first kind and what is a curve integral of 2nd kind? I have to understand this for my calculus exam.
I know how to compute the area of the wall between the graph of f(x,y) and curve $\gamma$ on XoY plane in this case:
The area is $\int f(x,y) \,ds$, where $\,ds = \sqrt {(\,dx)^2 + (\,dy)^2}$ and $x$ and $y$ get parametrized like this: $x = x(t)$ and $y = y(t)$
So my integral becomes (maybe not written 100% correctly): $\int f(x,y) \sqrt{(\,dx)^2 + (\,dy)^2}$
But we know that $x = x(t)$ and $y=y(t)$ the integral becomes: $\int f(x(t),y(t)) \sqrt{(\,dx)^2 + (\,dy)^2}$
Then: $\sqrt{(\,dx)^2 + (\,dy)^2} = \frac {\,dt}{\,dt} \sqrt{(\,dx)^2 + (\,dy)^2} = \frac {1}{\,dt}\sqrt{(\,dx)^2 + (\,dy)^2}\,dt = \sqrt{(\frac{\,dx}{\,dt})^2 + (\frac{\,dy}{\,dt})^2}\,dt$
And finaly the formula for the area: $\int f(x(t),y(t)) \sqrt{(\frac{\,dx(t)}{\,dt})^2 + (\frac{\,dy(t)}{\,dt})^2}\,dt$
Following this I have another question is this the first kind line integral? Is it the second kind line integral? Or it has nothing in common with first and second kind line integrals?
I played around with $f: R^2 \rightarrow R$ and $\gamma$ a parametrized path on XoY plane so a discussion with an $f: R^n \rightarrow R^m$ would be appreciated.