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Chapter 14: Q 32. (page 1154)
Use Green鈥檚 Theorem to evaluate the integral for the given vector field and curve.
and C is the circle with equation transversed counterclockwise.
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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 .
Work as an integral of force and distance: Find the work done in moving an object along the x-axis from the origin to if the force acting on the object at a given value of x is role="math" localid="1650297715748" .
Make a chart of all the new notation, definitions, and theorems in this section, including what each new thing means in terms you already understand.
Find the flux of the given vector field through a permeable membrane described by surface S.
, where S is the surface with the equation for .
If curl is constantly equal to on a smooth surface with a smooth boundary curve , then Stokes鈥� Theorem can reduce the integral for the surface area to a line integral. State this integral.
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