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Chapter 14: Q. 6 (page 1095)
What are the outputs of a vector field in ?
The outputs of a vector field in is is a three-dimensional vector.
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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.
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 .
, where S is the region of the plane with equation , where and , with n pointing upwards.
Use what you know about averages to propose a formula for the average rate of flux of a vector field F(x, y ,z) through a smooth surface S in the direction of n.
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