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. 17. F = 〈 x , y , z 〉; S is the upper half of the ellipsoid x 2 /4 + y 2 /9 + z 2 = 1.
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. 17. F = 〈 x , y , z 〉; S is the upper half of the ellipsoid x 2 /4 + y 2 /9 + z 2 = 1.
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.
17.F = 〈x, y, z〉; S is the upper half of the ellipsoid x2/4 + y2/9 + z2 = 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.
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 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 = ⟨y, z - x, -y⟩; S is the part of the paraboloidz = 2 - x2 - 2y2 that lies within the cylinder x2 + y2 = 1.
Firnd the area of the surface of the half cylinder {(r,0,z): r=6, 0s0S1, 0SzS5} using a parametric description of the surface.
Set up the integral for the surface area using the parameterization u=0 and v=z.
!!
S SO du dv
(Type an exact answers, using x as needed.)
The surface area is
(Type an exact answer, using x as needed.)
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