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Chapter 1: Q2P (page 4)
Is the cross product associative?
If so, prove it; if not, provide a counterexample (the simpler the better).
The values of the two cross product and , are not equal. Thus.
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Question: Check Corollary 1 by using the same function and boundary line as in Ex. 1.11, but integrating over the five faces of the cube in Fig. 1.35. The back of the cube is open.
Find the transformation matrix R that describes a rotation by 120掳 about an axis from the origin through the point (1, 1, 1). The rotation is clockwise as you look down the axis toward the origin.
Compute the line integral of
around the path shown in Fig. 1.50 (the points are labeled by their Cartesian coordinates).Do it either in cylindrical or in spherical coordinates. Check your answer, using Stokes' theorem. [Answer:3rr /2]
The height of a certain hill (in feet) is given by
Where y is the distance (in miles) north, x the distance east of South Hadley.
(a) Where is the top of hill located?
(b) How high is the hill?
(c) How steep is the slope (in feet per mile) at a point 1 mile north and one mileeast of South Hadley? In what direction is the slope steepest, at that point?
For Theorem 2, show that , , , and
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