Calculating Circulation of Field F w/ Stokes' Theorem

[mit_hacker]

Homework Statement

Use the surface integral in Stokes' theorem to calculate the circulation of field F around the curve C in the indicated direction.

(3) F = (y)i + (xz)j + (x^2)k.
C: Boundary of the triangle cut from the plane x+y+z=1 by the first octant, counterclockwise as seen from above.

Homework Equations

The Attempt at a Solution

I first calculated curl F which is (-x)i + (-2x)j + (z-1)k.
Moreover, r can be parameterized as x(i) + y(j) + (1-x-y)k.

The cross product of rx and ry will give i+j+k. The dot product of this and curl F will give -3x+z-1. Since dA = dxdy, I have to change z into x and y so this gives -4x-y.

From here, I do not know how to determine the limits. Can someone please tell me how to do so and if possible, provide suitable sites from where I can learn more on this?

Thank-you very much for the time and effort!

 

[HallsofIvy]

This is much like the other questions you have posted!

C: Boundary of the triangle cut from the plane x+y+z=1 by the first octant, counterclockwise as seen from above.

This question is asking you to integrate curl F over the surface having C as boundary: that is, the portion of the plane x+ y+ z= 1 in the first octant.
It should be obvious that region is a triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1). Projecting down to the xy-plane, just by ignoring the z coordinate, you get a triangle with vertices (1, 0), (0, 1), and (0, 0). That is just the triangle in the xy-plane bounded by the x-axis, the y-axis, and the line x+ y= 1. An obvious way to that is to take x from 0 to 1 and, for each x, take y from 0 to 1- x.

mit hacker, the thing that concerns me is that you should have learned this long before you were dealing with "surface integrals" and "Stoke's Theorem".

 

[mit_hacker]

Same here!

Dear HallsofIvy,

Thanks a lot for your help!

Yo are correct. I should have learned all of this before entering Multivariable Calculus but unfortunately, my course only allows for superficial learning an does not offer a deep appreication and understanding of the subject. For instance, we do not even do Lagrange multipliers, application of partial derivatives etc.. Forget all that. We don't even have to know the physical meaning of line integral etc... We are only taught how to apply them to solve questions.

To cover up for all that, I refer to MIT's OpencourseWare (hence my name!) but I'm finding it difficult to understand certain things and now, you've pointed out why that is so! So I guess read that up before going any further.

Thanks for the advice!

With best regards,
mit_hacker

 

[HallsofIvy]

Oho! I was wonderering if you were actually a student at M.I.T. (I will confess my first reaction was "he won't last long"!)

 

[mit_hacker]

No probs!

No problem! No hard feelings. Thanks a lot for your continued support!