220 CHAPTER 12. VECTOR VALUED FUNCTIONS
These pictures were drawn by maple. Note how they reveal both the direction and themagnitude of the vectors. However, if you try to draw these by hand, you will mainly wastetime.
12.3 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. 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
), com-
pute 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
220 CHAPTER 12. VECTOR VALUED FUNCTIONSThese pictures were drawn by maple. Note how they reveal both the direction and themagnitude of the vectors. However, if you try to draw these by hand, you will mainly wastetime.12.3. Exercises1. Here are some vector valued functions.Ff (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.kw BAN NNKOORRAN A BE PA AA ex LL BNVSNNN™SLfeer |= NANNA TP PAZ AAZ Moo LAITY NY NN™Li fe gir nNNAA WRN A UD AAA ww £ £dI|YNNN™Ll feels nn qr fr a a aa a ee eee ee ele wnsa a > oe || > ela | iVY NSN t]- 4 74 7 f ware ve ELAN NN Sni nn eee wee LS LIV NNN RANA AT TZ AANNN ole 44 A CL LLIAIVNNNN NANA TTF AAZAANe ee AF SSL L-RNNNNN NANA ART PT AAA: xy2. Find D(f) for f (x,y.z,) = (2, 6—37)?).3. Find D(f) for f (x,9,2) = (Gage. V4- (+P +2).4. For f (x,y,z) = (x,y,xy) 2 (x,y,z) = (y*, —x,z) and g (x,y,z) = (4,9z,27 — 1), com-pute the following.(a) fxg(b) 9x f(c) f-g(d) fxg:h(e) fx(gxh)(f) (f xg)-(gxh)5. Let f (x,y,z) = (y,z,x) and g (x,y,z) = (x7 +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(b) (kxg)xh(c) (f-g)xh(d) (f xg)-h
