Use Stokes' Theorem to Calculate F on Triangle 1,0,0...0,1,0...0,0,1

[joemama69]

Homework Statement

Use stokes theorem

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

Homework Equations

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

Then i found the gradient

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

 

[Shooting Star]

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


Since the surface S can be any surface, any convenient surface can be chosen. This is best done here by choosing the three triangles formed by the three positive axes and the lines joining the three given points.

Consider one such triangle -- for example, the one lying on the x-y plane, which is bounded by the x axis, y-axis and the segment joining (1,0,0) and (0,1,0). The unit normal on this surface is constant and the double integral can be found very easily, taking the proper limits for x and y. The other two can be done similarly (or just by inspection using symmetry property) and all the three then added to get the result.