# Stoke It

joemama69

## Homework Statement

Use stokes theorem

F = xyi + yzj + zxk on triangle 1,0,0,,,,,,,0,1,0,,,,,,0,0,1

## The Attempt at a Solution

First i found the curl F

curl F = -yi - zj - xk

Then i found the equation of the plane for the triangle

z = g(xy) = 1 - x - y

f(xyz) = z - g(xy) = z - 1 + x + y, grad f = i + j + k

so the integral will be dy from 0 to x-1, and dx from 0 to 1

(-yi - zj - xk) dot (i + j + k) = -y - z - x

integral (-y - z - x)dy = -y2/2 - zy - xy evaluated from 0 to x-1

integral -(x-1)2/2 - z(x-1) - x(x-1) dx

integral -x2/2 + x - 1/2 - zx + z - x2 + x dx evaluated from 0 to 2

-1/2 + 1 - z/2 + z - 1/2 = 2 - z/2 should that z be in there

$$\oint\limits_C {{\rm \bar F \bullet }d{\rm \bar r} = \int\limits_S {\int {(\bar \nabla \times } } } {\rm \bar F}) \bullet {\rm \bar n }\;d{\rm A}$$