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Chapter 14: Q. 20 (page 1119)
Give a formula for a normal vector to the surface S determined by y = g(x,z), where g(x,z) is a function with continuous partial derivatives.
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Give an example of a vector field whose orientation does not affect the outcome of Stokes鈥� Theorem.
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: The result of integrating a vector field over a surface is a vector.
(b) True or False: The result of integrating a function over a surface is a scalar.
(c) True or False: For a region R in the
(d) True or False: In computing , the direction of the normal vector is irrelevant.
(e) True or False: If f (x, y, z) is defined on an open region containing a smooth surface S, then measures the flow through S in the positive z direction determined by f (x, y, z).
(f) True or False: If F(x, y, z) is defined on an open region containing a smooth surface S , then measures the flow through S in the direction of n determined by the field F(x, y, z).
(g) True or False: In computing ,the direction of the normal vector is irrelevant.
(h) True or False: In computing ,with n pointing in the correct direction, we could use a scalar multiple of n, since the length will cancel in the term.
Integrate the given function over the accompanying surface in Exercises 27鈥�34.
, where S is the portion of the cone that lies within the sphere of radius 4 and centered at the origin.
Suppose that an electric field is given by
Compute the flux of the field through the unit cube .
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