# Questions about a proof of Stokes' theorem in my calculus 2 lecture notes

My lecture notes look to prove Stokes' theorem for the special case where a surface can be represented as the graph of some function, so $$z=f(x,y)$$.

The surface $$S$$ is parametrized as $$r(x,y)=(x,y,f(x,y))$$, where $$(x,y)$$ is in the region $$U$$ in the $$xy$$ plane. Now assume $$U$$ has a boundary curve $$C_u$$ and $$S$$ has a boundary curve $$C_s$$.

My first question is where it is said that the line integral of the vector field $$v=(v_1,v_2,v_3)$$ over the curve $$C_s$$ is equal to the line integral of the same vector field over the curve $$C_u$$. Why is this the case?

Lecture notes for my first question

My second question is I believe about a total derivative but I'm not sure. My lecturer has written that since $$z=f(x,y)$$, $$dz=(f_x)dx+(f_y)dy$$ where $$f_x$$ and $$f_y$$ denote the partial derivatives of $$f$$ with respect to $$x$$ and $$y$$ respectively and $$dx, dy, dz$$ denote normal differentials, not partial ones. Can someone also explain this equality to me, please?

Lecture notes for my second question

• @anomaly: thank you for correcting the spelling of Stokes Mar 20, 2020 at 16:53

The boundary $$\partial U$$ of the set $$U$$ in the $$(x,y)$$-plane has some parametrization $$\partial U:\quad t\mapsto\bigl(x(t),y(t)\bigr)\qquad(0\leq t\leq 1)\ ,$$ and the boundary $$\partial S$$ of your surface $$S\subset{\mathbb R}^3$$ then has the parametrization $$\partial S:\quad t\mapsto\bigl(x(t),y(t),f\bigl(x(t),y(t)\bigr)\bigr)\qquad(0\leq t\leq 1)\ .$$ Expanding the integrals $$\int_{\partial S}{\bf v}\cdot d{\bf r},\qquad \int_{\partial U}\bigl(v_1 dx+v_2 dy+v_3 dz\bigr)$$ with $$dx=x'\>dt,\quad dy=y'(t)dt,\quad dz=f_x\bigl(x(t),y(t)\bigr)\>x'(t)dt+f_y\bigl(x(t),y(t)\bigr)\> y'(t) dt$$ then shows that they have the same value.