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3. Consider the surface integral| (V × F) n dS, where F = (0, 0, ryz) and S is that portion of the paraboloid z = 1-r? - y² for z 20 oriented upward. Sketch the surface neatly and do the following. %3! (a) Evaluate the surface integral. [Do NOT use Stokes' theorem.] Jv x F) - nds = /]v × F\(z. y. S(z2, v)) V1+ (f-)? + (S»)° dzdy = Vg |V9| (V (b) Evaluate the surface integral in a simpler surface r + y? < 1, z = 0, oriented upward. This surface has the same boundary as the given paraboloid. Iv xF) - nds = (c) Use Stokes' theorem to verify the result in part (b). F dr =

3. Consider the surface integral| (V × F) n dS, where F = (0, 0, ryz) and S is that portion of the paraboloid z = 1-r? - y² for z 20 oriented upward. Sketch the surface neatly and do the following. %3! (a) Evaluate the surface integral. [Do NOT use Stokes' theorem.] Jv x F) - nds = /]v × F\(z. y. S(z2, v)) V1+ (f-)? + (S»)° dzdy = Vg |V9| (V (b) Evaluate the surface integral in a simpler surface r + y? < 1, z = 0, oriented upward. This surface has the same boundary as the given paraboloid. Iv xF) - nds = (c) Use Stokes' theorem to verify the result in part (b). F dr =

Advanced Engineering Mathematics
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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3. Consider the surface integral (V × F) - n dS, where F = (0, 0, xyz) and S is that portion %3D of the paraboloid z = 1-z? - y? for z 2 0 oriented upward. Sketch the surface neatly and do the following. %3D (a) Evaluate the surface integral. [Do NOT use Stokes' theorem.] (7 x F) nds = vx F)(z, y, f(7, 9) V1+ (f.)? + (f,)° dzdy Vg |Vg| (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. (V (c) Use Stokes' theorem to verify the result in part (b). F dr =
Transcribed Image Text:3. Consider the surface integral (V × F) - n dS, where F = (0, 0, xyz) and S is that portion %3D of the paraboloid z = 1-z? - y? for z 2 0 oriented upward. Sketch the surface neatly and do the following. %3D (a) Evaluate the surface integral. [Do NOT use Stokes' theorem.] (7 x F) nds = vx F)(z, y, f(7, 9) V1+ (f.)? + (f,)° dzdy Vg |Vg| (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. (V (c) Use Stokes' theorem to verify the result in part (b). F dr =
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