Stokes’ Theorem for evaluating surface integrals Evaluate the line integral in Stakes’ Theorem to determine the value of the surface integral ∬ S ( ∇ × F ) ⋅ n d S . Assume that n points in an upward direction. 20. F = 〈 x + y , y + z , z + x 〉; S is the titled disk enclosed by r ( t ) = 〈cos t , 2 sin t , 3 cos t 〉.
Stokes’ Theorem for evaluating surface integrals Evaluate the line integral in Stakes’ Theorem to determine the value of the surface integral ∬ S ( ∇ × F ) ⋅ n d S . Assume that n points in an upward direction. 20. F = 〈 x + y , y + z , z + x 〉; S is the titled disk enclosed by r ( t ) = 〈cos t , 2 sin t , 3 cos t 〉.
Solution Summary: The author explains Stokes' Theorem: Let S be an oriented surface with a piecewise-smooth closed boundary C whose orientation is consistent with that of S.
Stokes’ Theorem for evaluating surface integralsEvaluate the line integral in Stakes’ Theorem to determine the value of the surface integral
∬
S
(
∇
×
F
)
⋅
n
d
S
. Assume thatnpoints in an upward direction.
20.F = 〈x + y, y + z, z + x〉; S is the titled disk enclosed by r(t) = 〈cos t, 2 sin t,
3
cos
t
〉.
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 .
Find the surface area of the "Coolio McSchoolio" surface shown below using the formula:
SA = integral, integral D, ||ru * rv||dA
%3D
The parameterization of the surface is:
r(u,v) = vector brackets (uv, u + v, u - v) where u^2 + v^2 <= 1
A.) (pi/3)(6squareroot(6) - 8)
B.) (pi/3)(6squareroot(6) - 2squareroot(2))
C.) (pi/6)(2squareroot(3) - squareroot(2))
D.) (pi/6)(squareroot(6) - squareroot(2))
E.) (5pi/6)(6 - squareroot(2))
University Calculus: Early Transcendentals (3rd Edition)
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