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Chapter 14: Q. 9 (page 1106)
Write the first alternative form of from Example 3. (This alternative form is described immediately after Definition 14.4 and is used in Exercises 5–8.)
The alternative form of
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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’ 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’ Theorem can be interpreted as a generalization of Green’s Theorem.
(c) True or False: Stokes’ Theorem applies only to conservative vector fields.
(d) True or False: Stokes’ Theorem is always used as a way to evaluate difficult surface integrals.
(e) True or False: Stokes’ 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’ 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’ Theorem can be interpreted as a generalization of the Fundamental Theorem of Calculus.
(h) True or False: Stokes’ Theorem can be used to evaluate surface area .
Why do surface integrals of multivariate functions not include an n term, whereas surface integrals of vector fields do include this term?
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.
Consider the vector field . Find a vector field with the property that, for all points in role="math" localid="1650383268941" .
Evaluate the integrals in Exercises 43–46 directly or using Green’s Theorem.
, where R is the unit disk.What do you think about this solution?
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