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Chapter 14: Q. 1 (page 1134)
How might we generalize Green’s Theorem to two-dimensional regions that are surfaces in , rather than patches in the -plane? What sort of statement do you expect?
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Integrate the given function over the accompanying surface in Exercises 27–34., where S is the portion of the unit sphere in the first octant.
Find
Where S is the portion of the sphere with radius 2, centered at the origin, and that lies below the plane with equation , with n pointing outwards.
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 .
Use what you know about averages to propose a formula for the average rate of flux of a vector field F(x, y ,z) through a smooth surface S in the direction of n.
Compute a general formula for dS for any plane
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