3. Consider the surface integral / (Vx F)-n dS, where F = (0, 0, zyz) and S is that portion %3D of the paraboloid z = 1- r - y for z 20 oriented upward. Sketch the surface neatly and do the following. (a) Evaluate the surface integral. [Do NOT use Stokes' theorem. Vg (V /x F)(z,v.f(x, 9)) · V1+ (f.) + (f)? dzdy = (V x (b) Evaluate the surface integral in a simpler surface z + y? < 1, z = 0, oriented upward. This surface has the same boundary as the given paraboloid. %3D
3. Consider the surface integral / (Vx F)-n dS, where F = (0, 0, zyz) and S is that portion %3D of the paraboloid z = 1- r - y for z 20 oriented upward. Sketch the surface neatly and do the following. (a) Evaluate the surface integral. [Do NOT use Stokes' theorem. Vg (V /x F)(z,v.f(x, 9)) · V1+ (f.) + (f)? dzdy = (V x (b) Evaluate the surface integral in a simpler surface z + y? < 1, z = 0, oriented upward. This surface has the same boundary as the given paraboloid. %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3. Consider the surface integral // (VxF)-n dS, where F = (0, 0, zyz) and S is that portion
%3D
of the paraboloid z = 1- a? – y? for z 20 oriented upward. Sketch the surface neatly and do
the following.
(a) Evaluate the surface integral. [Do NOT use Stokes' theorem.
nds =
(V x F)(z, y, f(x, v)) .
Vg
1+(f-) +(f)? dzdy =
|V9|
F) 1
%3!
!!
(b) Evaluate the surface integral in a simpler surface + y? < 1, z = 0, oriented upward.
This surface has the same boundary as the given paraboloid.
vx F) nds =
(c) Use Stokes' theorem to verify the result in part (b).
F dr =
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