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. 21 . F = 〈 y , z − x , − y 〉 ; S is the part of the paraboloid z = 2 − x 2 − 2 y 2 that lies within the cylinder x 2 +y 2 =1.
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. 21 . F = 〈 y , z − x , − y 〉 ; S is the part of the paraboloid z = 2 − x 2 − 2 y 2 that lies within the cylinder x 2 +y 2 =1.
Solution Summary: The author evaluates the surface integral value by obtaining line integral in Stokes' theorem.
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.
21. F =
〈
y
,
z
−
x
,
−
y
〉
;
S is the part of the paraboloid z = 2 − x2 − 2y2 that lies within the cylinder x2+y2=1.
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.
Using Stokes’ Theorem to evaluate a surface integral Evaluate∫∫S (∇ x F) # n dS, where F = -y i + x j + z k, in the following cases.a. S is the part of the paraboloid z = 4 - x2 - 3y2 that lies within the paraboloid z = 3x2 + y2 (the blue surface as shown). Assume n pointsin the upward direction on S.b. S is the part of the paraboloid z = 3x2 + y2 that lies within the paraboloidz = 4 - x2 - 3y2, with n pointing in the upward direction on S.c. S is the surface in part (b), but n pointing in the downward direction on S.
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 .
Stokes’ Theorem for evaluating surface integrals Evaluatethe line integral in Stokes’ Theorem to determine the value of thesurface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upwarddirection.
F = r/ |r|; S is the paraboloid x = 9 - y2 - z2, for 0 ≤ x ≤ 9(excluding its base), and r = ⟨x, y, z⟩ .
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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