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Verify that Stokes' Theorem is true for the vector field F 6yzi – 3yj + rk and the surface S the part of the paraboloid z = 18 – x2 y? that lies abc the plane z = 2, oriented upwards. To verify Stokes' Theorem we will compute the expression on each side. First compute curl F - dS curl F = <0,1-6y Σ IL curl F· dS = Σ dy da where Y1 = 2pi 9 curl F· dS = Σ Now compute F. dr M M M M

Verify that Stokes' Theorem is true for the vector field F 6yzi – 3yj + rk and the surface S the part of the paraboloid z = 18 – x2 y? that lies abc the plane z = 2, oriented upwards. To verify Stokes' Theorem we will compute the expression on each side. First compute curl F - dS curl F = <0,1-6y Σ IL curl F· dS = Σ dy da where Y1 = 2pi 9 curl F· dS = Σ Now compute F. dr M M M M

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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Verify that Stokes' Theorem is true for the vector field F = 6yzi – 3yj + xk and the surface S the part of the paraboloid z = 18 – x2 – y? that lies above the plane z = 2, oriented upwards. To verify Stokes' Theorem we will compute the expression on each side. First compute curl F - dS curl F = <0,1-6y Σ 12 y2 curl F· dS = Σdy da where 2pi Σ Y2 = 9 Σ curl F· dS = Σ Now compute F. dr The boundary curve C of the surface S can be parametrized by: r(t) = (4 cos(t), Σ 0<t< 2n (Use the most natural parametrization) F. dr = Σ dt · dr = Σ M M M M W
Transcribed Image Text:Verify that Stokes' Theorem is true for the vector field F = 6yzi – 3yj + xk and the surface S the part of the paraboloid z = 18 – x2 – y? that lies above the plane z = 2, oriented upwards. To verify Stokes' Theorem we will compute the expression on each side. First compute curl F - dS curl F = <0,1-6y Σ 12 y2 curl F· dS = Σdy da where 2pi Σ Y2 = 9 Σ curl F· dS = Σ Now compute F. dr The boundary curve C of the surface S can be parametrized by: r(t) = (4 cos(t), Σ 0<t< 2n (Use the most natural parametrization) F. dr = Σ dt · dr = Σ M M M M W
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