15.11. EXERCISES 321
15.11 Exercises1. Here are some vector valued functions.
f (x,y) = (x,y) , g (x,y) = (−(y−1) ,x) , h(x,y) = (x,−y) .
Now here are the graphs of some vector fields. Match the function with the vectorfield.
-2 0 2-2
0
2
-2 0 2-2
0
2
-2 0 2-2
0
2
2. Find D(f) for f (x,y,z,w) =(
xyzw ,√
6− x2y2)
.
3. Find D(f) for f (x,y,z) =(
11+x2−y2 ,
√4− (x2 + y2 + z2)
).
4. For f (x,y,z) = (x,y,xy) ,h(x,y,z) =(y2,−x,z
)and
g (x,y,z) =(
1x,yz,x2 −1
), compute the following.
(a) f ×g
(b) g×f
(c) f ·g
(d) f ×g ·h(e) f×(g×h)
(f) (f ×g) · (g×h)
5. Let f (x,y,z) = (y,z,x) and g (x,y,z) =(x2 + y,z,x
). Find g ◦f (x,y,z).
6. Let f (x,y,z) = (x,z,yz) and g (x,y,z) =(x,y,x2 −1
). Find g ◦f (x,y,z).
7. For f,g,h vector valued functions and k, l scalar valued functions, which of thefollowing make sense?
(a) f ×g×h
(b) (k×g)×h
(c) (f ·g)×h
(d) (f ×g) ·h(e) l g· k(f) f×(g+h)
8. The Lotka Volterra system of differential equations, proposed in 1925 and 1926 byLotka and Volterra respectively, is intended to model the interaction of predators andprey. An example of this situation is that of wolves and moose living on Isle Royal
15.11. EXERCISES 32115.11 Exercises1. Here are some vector valued functions.f(y) = y), 99) = (—(v-1) x), h(x y) = (9).Now here are the graphs of some vector fields. Match the function with the vectorfield.2.7 7 2 2-7 7 1 TY YNpee NN VV USS ao ee issboyd zrt ~ rn tf 4 & “7 ¢ | Ns ™*\\N™N OT vp _ _ ~-~-— ~~ | NMOY \-- O--- ==- 0H - = ===CNN SOO? Hey yur ~nrnn ft +r -x-SSS TTT? -4 se LN RN aaaas20 git 7 FENN Gin tts s-2 0 2 -2 0 -2 0 22. Find D(f) for f (x, y,z,w) = (2. V6o—27y”).3. Find D(f) for f (x,932) = (Gaign V4-@ +? +2):4. For f (x,y,z) = (x,y,xy),R (x,y,z) = (y?, —x,z) and| 2g(x,y,Z) = =, YZ,x —1Xx, compute the following.(a) fxg (d) fxg-h(b) gx f (e) fx(gxh)(c) f-g (f) (f xg)-(gxh)5. Let f (x,y,z) = (y,z,x) and g (x,y,z) = (x? +y,z,x). Find go f (x,y,z).6. Let f (x,y,z) = (x,z,yz) and g (x,y,z) = (x,y,x? — 1). Find go f (x,y,z).7. For f,g,h vector valued functions and k,/ scalar valued functions, which of thefollowing make sense?(a) fxgxh (d) (f xg)-h(b) (kxg)xh (e) lg-k(c) (f-g)xh (f) fx(g+h)8. The Lotka Volterra system of differential equations, proposed in 1925 and 1926 byLotka and Volterra respectively, is intended to model the interaction of predators andprey. An example of this situation is that of wolves and moose living on Isle Royal
