Stokes’ Theorem for evaluating surface integrals Evaluate the line integral in stokes’ Theorem to determine the value of the surface integral ∬ s ( ∇ × F ) ∙ n dS . Assume n points in an upward direction. 22. F = 〈 4 x , − 8 z , 4 y 〉 ; S is the part of the paraboloid z = 1 − 2 x 2 – 3 y 2 that lies within the paraboloid z = 2 x 2 + y 2 .
Stokes’ Theorem for evaluating surface integrals Evaluate the line integral in stokes’ Theorem to determine the value of the surface integral ∬ s ( ∇ × F ) ∙ n dS . Assume n points in an upward direction. 22. F = 〈 4 x , − 8 z , 4 y 〉 ; S is the part of the paraboloid z = 1 − 2 x 2 – 3 y 2 that lies within the paraboloid z = 2 x 2 + y 2 .
Solution Summary: The author calculates the surface integral using Stokes' Theorem, where n is the unit vector normal to S determined by the orientation of S.
Stokes’ Theorem for evaluating surface integrals Evaluate the line integral in stokes’ Theorem to determine the value of the surface integral
∬
s
(
∇
×
F
)
∙ndS. Assume n points in an upward direction.
22.
F
=
〈
4
x
,
−
8
z
,
4
y
〉
; S is the part of the paraboloid z = 1 − 2x2 – 3y2 that lies within the paraboloid z = 2x2 + y2.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Stokes’ Theorem for evaluating surface integrals Evaluate the line integral in Stokes’ Theorem to determine the value of the surface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upward direction.
F = ⟨4x, -8z, 4y⟩; S is the part of the paraboloidz = 1 - 2x2 - 3y2 that lies within the paraboloid z = 2x2 + y2 .
Verify Stokes' theorem. Assume that the surface S is oriented upward. F=3zi−5xj+2yk; S that portion of the paraboloid z=36−x^2−y^2 for z≥0 I'm having trouble finding the normal n*dS in Stokes's Theorem
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