- #1

Felipe Lincoln

Gold Member

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## Homework Statement

With the stokes' theorem transform the integral ## \iint_\sigma \vec{\nabla}\times\vec{F}\cdot\vec{\mathrm{d}S} ## into a line integral and calculate.

## \vec{F}(x,y,z) = y\hat{i} -x^2\hat{j} +5\hat{k}##

##\sigma(u,v) = (u, v, 1-u^2)##

## v\geq0##, ##u\geq0##, ##u+v\leq1##

## Homework Equations

Stokes' Theorem

$$\oint_\gamma\vec{F}\cdot\vec{\mathrm{d}S} = \iint_\sigma\vec{\nabla}\times\vec{F}\cdot\vec{\mathrm{d}S}$$

## The Attempt at a Solution

The surface in this case is formed by a surface with several cuts: in the plane x+y=1, x=0 and y=0. So I have no idea on how to apply stokes' theorem here. I know this way: given a closed simple curve with not conservative vectorial field in it, the line integral of this vector field along this curve is equal to the integral of surface in which the surface is bounded by this curve. Thinking this way I can only imagine curves being bounds of a bounded surface, but here we got a surface that is part of a square and part of a curve. How to deal with this case?