Understanding Stoke's Theorem: Does Surface Integral Depend on Shape?

[nolanp2]

i'm trying to understand stoke's theorem and am having trouble seeing whether the surface integral for a given surface changes with any change in its shape, or if it only changes depending on the cross sectional area perpendicular to the direction of the vector field. can anybody help me out?

 

[benorin]

Stokes' Theorem equates the flux integral of the curl of a vector field $$\vec{F}$$ over an orientated piecewise-smooth surface $$S$$ to the line integral of $$\vec{F}$$ along the simple, closed, piecewise-smooth curve $$C$$ which is the boundary of the surface $$S$$, symbolically

$$\iint_S (\vec{\nabla}\times\vec{F})\cdot d\vec{S} = \oint_C \vec{F}\cdot d\vec{r}$$​

The value of the integral is fixed by the value of the line integral on the righthand side which could only change if the boundary curve $$C$$ changes, hence the integral is unchanged whether you integrate over the paraboliod $$S_1: z=4-x^2-y^2,z\geq 0$$ or the hemisphere $$S_2: x^2+y^2+z^2=4,z\geq 0$$ or the half cone $$S_3: (z-4)^2=4(x^2+y^2),0\leq z\leq 4$$ since these all have as their boundary curve $$C$$ the circle $$C:x^2+y^2=4$$.

 

[nolanp2]

perfect just what i wanted to hear! thanks