Vector Calculus Surface Integral
Could someone help me with these two questions:
1.) The ends of a circular cylinder of radius a and axis Oz are in planes z = 0, z = 2a. If S denotes the complete surface of the cylinder (including the end plates), evaluate Integral_S (z dS)
2.) Evaluate Integral_S (F.dS) where F = -4yz i - k and S is the surface x^2+y^2+z^2=1 where z>0
I've managed a few basic examples but I'm not sure how I go about these. I'd really appreciate it if someone could help me understand this. Thanks in advance.
1.) The ends of a circular cylinder of radius a and axis Oz are in planes z = 0, z = 2a. If S denotes the complete surface of the cylinder (including the end plates), evaluate Integral_S (z dS)
2.) Evaluate Integral_S (F.dS) where F = -4yz i - k and S is the surface x^2+y^2+z^2=1 where z>0
I've managed a few basic examples but I'm not sure how I go about these. I'd really appreciate it if someone could help me understand this. Thanks in advance.
Original post by Tanwir
Could someone help me with these two questions:
1.) The ends of a circular cylinder of radius a and axis Oz are in planes z = 0, z = 2a. If S denotes the complete surface of the cylinder (including the end plates), evaluate Integral_S (z dS)
2.) Evaluate Integral_S (F.dS) where F = -4yz i - k and S is the surface x^2+y^2+z^2=1 where z>0
I've managed a few basic examples but I'm not sure how I go about these. I'd really appreciate it if someone could help me understand this. Thanks in advance.
1.) The ends of a circular cylinder of radius a and axis Oz are in planes z = 0, z = 2a. If S denotes the complete surface of the cylinder (including the end plates), evaluate Integral_S (z dS)
2.) Evaluate Integral_S (F.dS) where F = -4yz i - k and S is the surface x^2+y^2+z^2=1 where z>0
I've managed a few basic examples but I'm not sure how I go about these. I'd really appreciate it if someone could help me understand this. Thanks in advance.
With the first one, I think you can cheat using the divergence theorem. However, to perform a surface integral, you generally split it up into a bunch of easy surfaces (like the two end plates and the central column), and then integrate over those, remembering to be careful about the orientation of the surface. (I detest vector calculus with a fiery passion, though, so take anything I say with a pinch of salt.)
First one: change the surface integral into three separate double integrals (over the two plates and the column). Change coordinates as appropriate to make the integrals nicer.
Second one: I'd definitely use the divergence theorem - I think that's probably what the question is about.
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