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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鈥�8.)
The alternative form of
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Use the curl form of Green鈥檚 Theorem to write the line integral of F(x, y) about the unit circle as a double integral. Do not evaluate the integral.
Compute dS for your parametrization in Exercise 9.
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 .
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.
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