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Chapter 14: Q. 16 (page 1140)
Given an integral of the form what considerations would lead you to evaluate the integral with Stokes鈥� Theorem?
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Compute n for the surface S in Exercise 12.
Find the flux of the given vector field through a permeable membrane described by surface S.
, where S is the paraboloid with equation role="math" localid="1650299986090" that lies above the annulus determined by in the XY-plane.
Q. True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: Stokes鈥� Theorem asserts that the flux of a vector field through a smooth surface with a smooth boundary is equal to the line integral of this field about the boundary of the surface.
(b) True or False: Stokes鈥� Theorem can be interpreted as a generalization of Green鈥檚 Theorem.
(c) True or False: Stokes鈥� Theorem applies only to conservative vector fields.
(d) True or False: Stokes鈥� Theorem is always used as a way to evaluate difficult surface integrals.
(e) True or False: Stokes鈥� Theorem can be interpreted as a generalization of the Fundamental Theorem of Line Integrals.
(f) True or False: If F(x, y ,z) is a conservative vector field, then Stokes鈥� Theorem and Theorem 14.12 together give an alternative proof of the Fundamental Theorem of Line Integrals for simple closed curves.
(g) True or False: Stokes鈥� Theorem can be interpreted as a generalization of the Fundamental Theorem of Calculus.
(h) True or False: Stokes鈥� Theorem can be used to evaluate surface area .
Find the integral of on the portion of the plane with the equation
with 2 鈮� x 鈮� 7 and 1 鈮� z 鈮� 2.
Find the area of S is the portion of the plane with equation y鈭抸 =
that lies above the rectangle determined by 0 鈮� x 鈮� 4 and 3 鈮� y 鈮� 6.
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