Choosing a more convenient surface The goal is to evaluate A = ∬ S ( ∇ × F ) ⋅ n d S , where F = 〈 yz , – xz, xy〉 and S is the surface of the upper half of the ellipsoid x 2 + y 2 + 8 z 2 = 1( z ≥ 0). a. Evaluate a surface integral over a more convenient surface to find the value of A . b. Evaluate A using a line integral.
Choosing a more convenient surface The goal is to evaluate A = ∬ S ( ∇ × F ) ⋅ n d S , where F = 〈 yz , – xz, xy〉 and S is the surface of the upper half of the ellipsoid x 2 + y 2 + 8 z 2 = 1( z ≥ 0). a. Evaluate a surface integral over a more convenient surface to find the value of A . b. Evaluate A using a line integral.
Solution Summary: The author explains the Stokes' Theorem: Let S be an oriented surface in R3 with a piecewise-smooth closed boundary C.
Choosing a more convenient surface The goal is to evaluate
A
=
∬
S
(
∇
×
F
)
⋅
n
d
S
, where F = 〈yz, –xz, xy〉 and S is the surface of the upper half of the ellipsoid x2+ y2 + 8z2 = 1(z ≥ 0).
a. Evaluate a surface integral over a more convenient surface to find the value of A.
b. Evaluate A using a line integral.
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.
Find the area of the surface generated by revolving the curve y = 2x -x , 0.5sxs1, about the x-axis.
The area of the surface generated by revolving the curve y = 2x -x, 0.5sxs1, about the x-axis is
square units.
(Type an exact answer, using t as needed.)
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