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Chapter 14: Q 6 (page 1153)
Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.
The vector field is non conservative
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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 .
Find the integral of on the portion of the unit sphere that lies in the first octant, above the rectangle in the XY-plane.
What are the outputs of a vector field in ?
Find the area of S is the portion of the plane with equation x = y + z that lies above the region in the xy-plane that is bounded by y = x, y = 5, y = 10, and the y-axis.
Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) Two different surfaces with the same area. (Try to make these very different, not just shifted copies of each other.)
(b) Let S be the surface parametrized by
Give two different unit normal vectors to S at the point
(c) A smooth surface that can be smoothly parametrized as
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