surface integral question

[Matt R.]

I have a mathematical physics final tomorrow, and I'm having trouble with surface integrals. I can't reach my prof through email, so I was hoping someone here could help me. Here is an example question:

If F = 2y i - z j + x^2 k and S is the surface of the parabolic cylinder y^2 = 8x in the first octant bounded by the planes y = 4 and z = 6, evaluate:

|| F.n dS

The || is just a double integral over the surface S, and F.n is the force vector dotted with the normal.

Essentially what I am stuck on is expressing the given surface as a vector function so I can dot it with the force. Or alternatively, finding the normal to such a surface to get F.n and finding dS. I don't have any trouble if I am given the surface expressed as a vector function, but I don't know how to convert.

Thanks for any help in advance.

[MadTiger]

http://en.wikipedia.org/wiki/Surface_integral
http://tutorial.math.lamar.edu/AllBr...ectorField.asp

That guy's notes has a problem like yours.

[sweetjazz]

You can parameterize the surface via r = (y^2/8, y, z). Taking the partials with respect to y and z give tangent vectors to the surface, r_y = (y/4, 1, 0) and r_z = (0, 0, 1). A normal to the surface can be found by taking their cross product, n = r_y x r_z = (1, -y/4, 0). This normal is the one with the right length to account for the change in area induced by the parameterization. This normal points outward.

F dot n dS = (2y, -z, y^4/64) dot (1, -y/4, 0) = 2y + yz/4 dy dz

Hope that helps.

[Matt R.]

Yes, that's really helpful. The idea of parameterization is what's throwing me off. I don't understand how to get the equations in all cases. I wish I would have payed attention in my multivariate calc course [img]/images/graemlins/blush.gif[/img].

But, that answer helped -- I see where you got the vector and hopefully I won't get thrown off by some weird surface on the final.

Thanks.