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Chapter 14: Q. 47 (page 1120)
Find
and S is the portion of the hyperboloid that lies between the planes
z = 鈭�4 and z = 0, with n pointing outwards.
Ans:
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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 .
What is the difference between the graphs of
Given a smooth parametrization for a 鈥済eneralized cylinder鈥� S, given by extending the curve y = x2 upwards and downwards from z =鈭�2 to z = 3.
Give an example of a vector field whose orientation does not affect the outcome of Stokes鈥� Theorem.
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