1. No, f(x,y) is not linear because the increments of f for the
same changes in x and y is not the same in either the rows or the columns.
2. False. If
and
and perpendicular vectors, the
angle between them is
.
The dot product is defined as
3. True.
so
for all .
4.Since and for , we can make the following table:
t | x | y |
0 | 0 | 1 |
1 | 0 | |
0 | -1 |
which shows the answer is c.
5. Because f(x,y) depends only on x, the contours on the
xy-plane will be straight lines up and down, parallel to the y-axis
as shown in Figure a. Constant x gives constant z.
6. The effect of the vector field on the top (y > 0) is larger and
in the direction of the orientation of the curve, than the effect on
the bottom (y<0), which is opposite to the orientation of the curve.
7. Using the curl test,
.
So
is not conservative.
8. True, the functions are just multiples of each other.
9. Since PV = RT, so,
a.
and
b. For the given conditions:
so
10.
a. The equation of the line is
b. We know
x2 + y2 = r2 so this suggests polar coordinates:
By u-substitution, with u=-r2 get
11.
a. Stokes' Theorem says that the circulation of a vector field around
the closed boundary of a capping surface is the same as the flux
through the surface. In symbols:
where:
b. In
12. To be supplied.
13.
a.
so
and
And these add to 0.
b. Since the sphere is of radius 1, and centered at (0,0,2), the
Divergence Theorem can be used, since the field
is defined
everywhere within the sphere. Thus
14.
a.
b. The function f increases most rapidly in the direction and the rate of change is 0 along the contour, in the directions .