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. 24. F = 〈 e x , 1 / z , y 〉 ; S is the part of the surface z = 4 − 3 y 2 that lies within the paraboloid z = 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. 24. F = 〈 e x , 1 / z , y 〉 ; S is the part of the surface z = 4 − 3 y 2 that lies within the paraboloid z = x 2 + y 2 .
Solution Summary: The author evaluates the surface integral by obtaining line integral in Stokes' theorem, where n is the unit vector normal to S determined by its orientation.
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.
24.
F
=
〈
e
x
,
1
/
z
,
y
〉
; S is the part of the surface z = 4 − 3y2 that lies within the paraboloid z = x2 + 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.
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.
Let F=xi+yj+zk and let f(x,y,z)=x^2e^(y-z).
a) Describe a surface S together with orientation so that double integral (F.dS)<0
b) Explain why you cannot find a surface S so that double integral f(x,y,z)dS<0.
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 .
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