?reas,
A d v a n c e d Calculus N o t e s 1 4 . 1 A B C
Vector-Valued Function:
Name
H b y
Oy B e
.
(as studied in chapters 10 and 11)
A v e c t o r - v a l u e d f u n c t i o n , also r e f e r r e d t o as a v e c t o r f u n c t i o n , is a m a t h e m a t i c a l f u n c t i o n o f o n e o r m o r e variables w h o s e
r a n g e is a set o f m u l t i d i m e n s i o n a l v e c t o r s o r i n f i n i t e - d i m e n s i o n a l vectors. The input o f a vector-valued f u n c t i o n could be a
scalar o r a v e c t o r .
A line or curve in space is an example o f
a
vector valued function whose input
is
some parameter
¢
and whose output are
vectors whose tips, when connected, form the line o r curve in space.
Vector Field:
A v e c t o r f i e l d is a t y p e o f v e c t o r - v a l u e d f u n c t i o n . tn v e c t o r calculus, a v e c t o r f i e l d is an assignment o f a v e c t o r t o each
p o i n t in a subset o f Euclidean space. A v e c t o r field in t h e plane, f o r instance, can be visualized as a collection o f a r r o w s
w i t h a given m a g n i t u d e and d i r e c t i o n each attached t o a point i n t h e plane. V e c t o r fields are o f t e n used t o m o d e l , f o r
e x a m p l e , t h e s p e e d a n d d i r e c t i o n o f a m o v i n g fluid t h r o u g h o u t space, o r t h e strength and direction of some force, such as
t h e m a g n e t i c o r g r a v i t a t i o n a l f o r c e , as it changes f r o m p o i n t to point.
in c o o r d i n a t e s , a v e c t o r f i e l d o n a d o m a i n in n-dimensional Euclidean space can be represented as a vector-valued
f u n c t i o n t h a t associates an n-tuple o f real n u m b e r s t o each point o f t h e domain. This r e p r e s e n t a t i o n o f a v e c t o r f i e l d
d e p e n d s o n t h e c o o r d i n a t e system, and t h e r e is a well-defined t r a n s f o r m a t i o n law in passing f r o m o n e c o o r d i n a t e system
t o t h e o t h e r . V e c t o r fields are o f t e n discussed on open subsets o f Euclidean space, but also make sense on o t h e r subsets
such as surfaces, w h e r e t h e y associate an a r r o w t a n g e n t t o t h e surface a t each p o i n t (a t a n g e n t v e c t o r ) .
A slope field, as studied in BC Calculus, is a v e c t o r field t h a t assigns to each p o i n t
a
v e c t o r r e p r e s e n t i n g t h e slope o f t h e
c u r v e at t h a t p o i n t .
The g r a d i e n t v e c t o r f u n c t i o n is an example o f a v e c t o r fieid t h a t w e have already studied. The gradient v e c t o r f u n c t i o n
assigns a v e c t o r t o each p o i n t in 2D o r 3D space. That v e c t o r is composed o f t h e partial derivatives a n d has m e a n i n g w i t h
r e g a r d t o d i r e c t i o n o f m o v e m e n t a n d m a g n i t u d e o f greatest increase in t h e f u n c t i o n at t h e given p o i n t .
A v e l o c i t y v e c t o r f i e l d describes t h e m o t i o n s of systems o f particles in space.
A g r a v i t a t i o n a l v e c t o r f i e l d describes t h e f o r c e o f a t t r a c t i o n exerted on o n e particle by a n o t h e r particle.
Examples o f pictures o f v e c t o r fields:
a.
2
dimensional slope field of the function f ( x , y)=x+y
e
2
the 2 dimensional
gradient v e c t o r f i e l d
V f ( x , y ) = ( 2 x , 2 y)
associated with curve
f ( x , y ) = x? +y?
. Shown
are b o t h the v e c t o r field and s o m e o f the contours o f the function.
NNN NAVAL
PACER
N N NNO
E
NNN
P
NNN NUM T
e
S A N NS U t
~
www
IP
sitimissys
oy
3
m
w
o
e
?
k .
e
o s e
C
r
C S S
S S L
c.
P L E C s e
P S Y Y
R s
e
e x e
ae, < P e e
AR K A
PD
e
d
A V
az
The 3 dimensional vector field
F(x, y,z)= 2xi
d. Velocity vector field formed by a rotating wheel.
e. Velocity v e c t o r field f o r m e d by air f l o w a r o u n d a m o v i n g object.
« gravitational vector field that describes the force of attraction exerted on one particle with mass
mm,
located at
p o i n t (x, y , 2 ) by another particle with mass 7 , located at the origin.
zz
~ wy
;
.
@&y,2
WV
, yt
?\
Nw
~
?Samy
k
m.,i s b o c aaf( nx , ye , 2d) .
a n , i s l o c a at te( 0d, 0, 0).
aa
e
Since a v e c t o r field consists o f i n f i n i t e l y m a n y vectors, it's impossible t o s h o w t h e m all. T h e r e f o r e , w h e n i l l u s t r a t i n g a
v e c t o r f i e l d , w e will o n l y d r a w a r e p r e s e n t a t i v e sample.
W e m i g h t c o n s i d e r d r a w i n g a set o f v e c t o r s w i t h a specific
relationship.
a.
It m i g h t be useful t o consider d r a w i n g a f e w sets o f vectors w i t h t h e same m a g n i t u d e . For e x a m p l e , i f a
v e c t o r f i e l d is r e p r e s e n t e d by
F ( x , y ) = a@i+ y j , d r a wa
set o f vectors w i t h m a g n i t u d e ( F | = k
or
k = V a ° +b?
b.
If asked t o d r a w a v e l o c i t y v e c t o r field, you will naturally be limited t o d r a w i n g v e c t o r s at t h e p o i n t s
l o c a t e d on t h e m o v i n g object.
Ex1: Draw the associated vector field and describe any general qualities of the vector field that you notice:
a.
b. F ( x , y ) = - i + j
F(x,y)=.3i-4j
At every p o u t in space, the vector 4 3 - 4 )
Should be d r a w n .
Shuld be d r a w n ,
Bachvects hoy Macnitide
\ F ] = 937 4.47
=
ye O44.1e
7
y.ld
?
bach vectr
us
At every pont in 2s pace, He vector 1 , 1 7
pracuiiel
wh
G3-4> 7°?
ws
Each vechrhav Magnitude [ F i z V i s =t 02
Each veche
1s
poraliel t 0 € 1 , 1 9
a
_ B(xy)= v i t
Mt every
e
x
At every pointin @spwe , drow vechr €24,7 7
pomt in Bspace , draw vester Ly, ¥.>
sana"
F(x, y)=2xi+yj
d.
S a v e s bs wagntade (
:
boon
Gacnvector has Magnitude
JF
l= Vytex? = K
x2ty? = 2
This aeons t h a t a t all p o t s
m e r c e
x2ry2sk2) t h e veehr drawn will
im t e n g i n t o t h e
be
This
equal
means t a t cé-all powb on ellipse
Elhpse
?
,
?
?
.
f o
a t pont(0,-t)
r=lond
Vechr <=} 0 D
Atevery pant iaZspace, d w n
<y,-3?
yO
|
f.
:
Each
<0,.57
at (U3)-9
3, 5)
(0,0) ?-
<o.s5)
(a,3:)-5
43,65)
Bo)
©,°5)
(3,4)
43,5)
(01) >
?4.5?
(tl)
<4,.5)
5 ?
(04) ~
3.5)
a
(34)
longer vectw
-
45)
€-1, 65)
g. e l e s y . 2 ) = - i 4 4
F(x,y,z)=k
At every p a n t m d s p a c e , drawZo
7
2)
=D e B ?
V e d £0,127
r A fl o oeA
COU L o
at (1,0) >
y e t
3
a t p o w t(1,0), c=!
E a c h v e c t y h a s Magnitude
yrt.25
Gorter. Ahpses h a e
at pant (2,0)
hese?
F(x,y)=yit+.5j
e.
Sieh
4
_
IF \ =
o r
we
a
xhey"=4
e7!
the veckr drawn will have the same length.
Circles
|
x
rodus of-thatercle
a n d b e ine d i r e c t i o n < y , x >
e r y =
F l e FH
a a y =k
2
Fxraxy? = k
vector hav l e n g t h 2
7
A t every p o u t in 3 Space ,dmwd-',3
Eacnvecte has l e a g h T a
F 5 7 2 ) =2k
_
very pout an 3 space, draw <2, 0 , x )
Fac
veo
hay beagth \el= x
i.
V(x, y,2) =(16-x? - y ? ) k where x? + y? <16
At panb
ot
inside
x*ty*=I6,
QS
ts zem onthecircle
yoo go t o w o d l t h e o r t g i n .
a t ( 4 0 , 0 ) ?» { 0 , 0 , 0 9
(0,9,4) ?
<0,0,07
(39,0
~~?
¢0,9,7 9
( R o , 0 >)
<00,127
(1,00) 9
<0,0,85)
(000) >
(00,16)
(91) >
(6,015)
Ex. 2 M a t c h t h e v e c t o r fields w i t h t h e i r plots.
a.
F(x,y)=(y,x)
~
x
?x
x
?
x
x
~~
a
x
c.
F(x, y)=(x-2,x4+1)
a e
d.
T S
F(y)=(v4)
fos
~s
T Y
*
£ 0 , 0 ,
i p - x * - yD*
Bak vcelr ha keaglh Ifl= f o r o s ( e r y
t h e lead,ofthe vector
i
draw
are
t e x ?
get
locgee
watch the vector fields with their plots.
F(x,y,z)=(1,2,3)
a.
(all passe vectors )
(Ww)
b.
F( x , y , z ) = ( 1 , 2 , 2 )
C o r m e xy plane,p o t o p
?below
yp l o r e / p o r t
down)
o~
td)
c.
F(x, y,z)=(x,y,3)
( a ! pond oy b a t wayne ed duccha® )
iy
d.
F ( x , y , z ) = ( x , y ,z )
( s e a ,
Y)
f u n n
ove
y o )
« t e n t V e c t o r Fields:
called a gradient vector field.
We know that the gradient function can be used to form one type of vector field
|
V i ( x y) = £ 0 , y i + £(x, y ) j
Therefore V / is really
a vector field: on R? and is called agradi
ikewi
.
eg:
gradient vector field. Likewise,
if f i s a scalar function of three variables, its gradient is a vector field on R° given by
V i f (x, y , z) = F ( x , y, z i + F ( x , y, 2 4 + f e y, z ) k
Ex. 4 Find, but do not draw, the gradient vector field for each function.
a.
be
Fy
f ( x , y ) =xe?
xety + e y
x e%
=e¢
b.
?Ky +1)
Sf
(x,y) = t a n ( 3 x - 4 y )
f (x,y,z) =xIn(y?-2z)
c.
f , =3sa*(3x-Hy)
=
£
f = ?tsa:?(3x-4y>
= tetY
oo j
xy
Ke
Uv
¥i)
a a
y-22
me
F(x, Y e e ?Gy
(2%
4 ty-22)
F o n :? ( a t o
( Fay )
) )
S e e -*
?
=
£
a e
cLiniy.aa), p- > = 3 x >
e e e
C o n s e r v a t i v e V e c t o r Fields:
scalar function, that is, if there exists
.
_¥@
?y-ae
#87
-s
for F
=
a
oaee
e
e
o t a l
v e c t o r field F is called a conservative vector field if it is the gradient o f some
function f
such that F = V f
.
Inthis situation f
is called the potential function
Not all vector fields are conservative, but such fields do arise frequently in physics. In example 4 above, we found
the conservative vector fields for the potential functions f ( x , y ) o r ( x , y , z )
7
os.
_
e r
n
a
e
t
A
>
D E F I N I T I O N OF CONSERVATIVE VECTOR F I E L D
y
e y e
A v e c t o r f i e l d F is c a l l e d c o n s e r v a t i v e i f there exists a d i f f e r e n t i a b l e f u n c t i o n
J such that F = Vf. The function f i s called the potential function for F.
?Let.M and N have continuous first partial derivatives onS a p a
ISK
s
, v
7
ce
{
R.
vector field given by F(x, y) = M i + Nj is conservative i f and only i f
p o d
=
.
,
d e c d
V o l a
tow
of
em
,
e
j e
cf
d e n v
¥
tuft
a
r
of
f i a t
p
t h
P e e . + +23
4
This t h e o r e m is based o n t h e u n d e r s t a n d i n g t h a t antiderivatives are needed t o d e t e r m i n e i f t h e c o m p o n e n t s o f t h e v e c t o r
field in f r o n t o f y o u r e p r e s e n t t h e partial derivatives o f t h e s a m e p o t e n t i a l f u n c t i o n .
Since
F(xy
M i
~hiy
t y .
M= f
l
r
)
Fay n y )
a
ee
=OM
ee
and
ee,
N e Fy i g ) ? > Fyx (?4 )? a N
av
But
g
°
>Fyy
Fx
t
OM
*
. aN
ay
ay
Clatauts
The
<termine whether each vector field is conservative. If it is, find the potential function.
F(x,y)=2y'i+3y'"j
a.
S>
fy
axy?
Be =
= x?
c e
ey
.
Yer and
b.
£
o e Bye
any?
z=
By*x?
>
( s y )
=
z+ dys?
a
foey)
=Hyhek
( 0 1 - 2 )
=
oe
equialenty
8(4fr?)» = o(ane)
=x?y?
4
ed
= by
On
apy
/
)
y t ,
- A x y ? >
oR.
fey?
fy=-auy?
2 ey!
ge = Oxy?
aR
oy
Zz: vy?
=P
= d(y)
d(-avy*)
*
Y
- ay?
z= ax ye
=
?ly?
(e)
Qe)
c.
f
F(x, y ) = e * (cos yi+sin yj)
.
a
Ccoy
ae
ox
¥
=e
etfs
?iny
f -
ef
=€
an
TID
etsy)
Ox
5
-
.
Cosy
Z = easy
oe
ay
(ze
2
-
e" sing
Seay
etsiny
=
Cue)
e*siny
e'avy )
oy
sical I n t e r p r e t a t i o n o f t h e Curl o f a V e c t o r Field
The curl o f a v e c t o r field measures t h e t e n d e n c y f o r t h e v e c t o r field t o s w i r l a r o u n d . Imagine t h a t t h e v e c t o r f i e l d
represents t h e v e l o c i t y v e c t o r s o f w a t e r in a lake. If t h e v e c t o r f i e l d swirls a r o u n d , t h e n w h e n w e stick a paddle w h e e l into
the water, it w i l l t e n d t o spin. The a m o u n t o f t h e spin w i l l d e p e n d o n how w e o r i e n t t h e paddle. Thus, w e should e x p e c t
t h e curl to be v e c t o r valued.
To be technical, curl is a vector, w h i c h means it has a both a magnitude and a
direction. The magnitude is simply the amount of twisting force at a point.
The direction is a little more tricky: it's the orientation o f the axis of y o u r
paddlewheel i n order to get maximum rotation. In other words, it is the direction
which w i l l give you the most ?free work" from the field. Imagine p u t t i n g y o u r
paddlewheel sideways in the whirlpool - it wouldn't turn at all. I f you put it i n the
proper direction, i t begins turning.
The fact t h a t there is a ?twist? means the field is n o t c o n s e r v a t i v e (this has n o t h i n g
to do w i t h its political views).
A conservative field is ?fair? in the sense that work needed to move f r o m p o i n t A to
p o i n t B, along any path, is the same. For example, consider a river: its field is
conservative. Sure, you can get
a
free ride downstream, b u t t h e n you have to do
w o r k to get back to your starting point. Or, you can do w o r k to move upstream, and
get a free ride back. Either way, the amount o f work you "put in" is the same as w h a t
you get back.
However, i n a field w i t h curl (like
a
whirlpool), you can get a free ride by moving i n
t h e d i r e c t i o n o f the twist. I n a w h i r l p o o l , you can get a free t r i p by moving w i t h the
current
ina
circle. I f you fight the current and go the wrong way, you have to use
energy w i t h no free ride at all.
Conservative fields have zero curl: there are no free twists to push you along.
Alternatively, i f a field has curl, it is not conservative.
For example, c o n s i d e r this v e c t o r field:
F
=
[ - y , z , 0}
It l o o k s l i k e t h i s :
B E E
B
we H E A R
6
K R K
R r w HE t K
KE
K
crew eewKSK
o
cf
.
r
e
W O N
O
-
4
t e
we
-
y+
N
?
w e
ff
O S
2
s=
e
t e Ame r
ww O f
e
T
r
%,
N
Thecurl looks
V x F
=(
like this:
4he d r e n
0 , 02,]
T h i s means that anywhere vou stand in this field, you w i l l rotate at
around the z-axis. I n physics it would have
a
a
rate o f ?2?
more strongly defined meaning,
like 2rpni o r something, but that?s not important.
T h i s vector field is actually quite interesting for d e m o n s t r a t i n g curl, because o f
the fact that its curl is constant. As you can see, the vectors are stronger at the
edges. However, because the field just completes a simple circle, the outer edges
rotate at the same rate as the i n n e r edges, meaning that you can stand anywhere
i n the field and will undergo the same a m o u n t of rotation in a given time.
You s h o u l d also note t h a t the curl is a measure of h o w much you w i l l be made to
rotate a i f i a t p o i n t ,
consider the following vector field:
F=
| -ysin(2}oo0(y), 0
]
I t looks like this:
?
??
?
'
;
;
}
?
4
?
f
x
x
x.
~
o2
9
2
4
Its curl is this:
V x F = [ o , 0, - y s i n ( z ) s i n ( y ) + s i n ( z ) c o s ( y ) +3 ]
N o w v o u r rate o f r o t a t i o n (still just a r o u n d the z-axis) depends o n where y o u
stand. This is obvious by looking at the vector plot above. The d i r e c t i o n o f the
vectors are a lot more haphazard, so clearly your r o t a t i o n rate w o n ' t be u n i f o r m .
In fact, i f you stand at the point ( c = o n , y = 2 7 ) , there w o n ' t be any curl at
all, even t h o u g h you still move i n the vector field. T r y it:
For the curl:
V x F = [o,9, ?
=
2
n t sn ) i s i n ( 2(7 ) + s i n ( m ) c o s ( 2 n ) + 3
[0, 0,0}
The curl is zero e v e r y w h e r e in each o f t h e f o l l o w i n g v e c t o r fields.
W e will see the g e o m e t r i c i n t e r p r e t a t i o n of Curl later in the chapter. For now. , w e n e e d it to use t h e t h e o r e m b e l o w t o
test f o r conservative v e c t o r fields in space.
FINITION
I f F(x, y,2) = f , y. D i + 8G, y , 2 ) 5 + kG, y, 2 k , then we de-
15.1.5 DE
fine the curl o f F, written curl F, to be the vector field given by
_(d8_ 9),
(38 of
alk = ( 2 - H ) i + (F( 9 ) i5,
+ (2¢_
>)
k
(8)
curl F has vector values (i.e., curl F is itself a vector field). Moreover, for computational
purposes it is useful to note that the formula for the curl can be expressed in the determinant
form
curl F =
THEOREM 15.2 TEST FOR CONSERVATIVE
VECTOR
FIELD IN SPACE
Suppose that M, N, and P have continuous first partial derivatives in an open
sphere Q in space. The vector field given by F(x, y, 2) = Mi + Nj + Pkis
conservative if and only if
curl F(x, y, z) = 0.
That is, F is conservative i f and only i f
ap
_
aN
ax
az?
dy
aM
@P_
|,
a _aM
N
ax
az?
ay?
Ex. 6 Find t h e curl of the v e c t o r field F at the indicated point.
a.
F ( x , y , 2 z ) = 2 x ° z i ? 2 x yzzj +
k at ( 2 , - 1 , 3 )
curl >
a
4
dx
dy
F(x, y
b.
?
,
¢
a
j o
a
&
y
xe
z e
>
(
zea)
)
Leta,
Bean,
z s i)n y =
i - e *e c o s* y j at ( 0 , 0 , 3 )
Cucl - | a d
dz
ox
[e*smy
x72 -2x2 y e y
=
=
,
.
a
s f
x2
-4
o-x*
4 (
x )
2
B )
7
\
fe
o e s
}
i
oa
oy
Ac
k (-az-0
r k /
}
a
gor]
-
=
7
t(o)
2.
ey
ae
e'cay
oO
a
%
.
|
?erary
o y -e>}
2 % ) bea) = e 8? |
7k
2.
2.
i
]
a
sity o
Oo
?
a
oe
+
S
ein
2%
x
>
§( 0 )
a
k ( ~Ccary ? e * cosy)
=
<4
2, ~aeeayy>
loos)
122, 2 7
?
,
e w h e t h e r each v e c t o r f i e l d is c o n s e r v a t i v e .
If i t is, find t h e p o t e n t i a l f u n c t i o n .
em iri
p(ane)oe ( i t s i t k )
z } i
59k
4
2
2
2
a
( | 8
=
|
i / #
/ &
BY
2
5|
A
OK €
ny
i(o-de) -
j(o-ye}
«
U e€)
(-e ¥) 1 , 0 7 # 0
=
hot consentarhue
vectorf
l l )
e e e
b.
find
curl
F(x, y,z) =3x"y"zit 2 x yzjt+ x y k
ie
|
=
2 2
2 [
x
Me
2
R l - i ] e
e l k
ey
aye o P
tae
i
l
y
*
we
de?ye
C
(avy
arty)
~
J ( B Y = 94 *)
w
? This
=O
f.
to =
ay? y 2
%
fe d a y s
fz
f e a x yr
r y
-
?o fr s= 3x4y 22
5
oo Ff x y
2
Oz
d y s
fz
eye
qa
a k (Ox'ye - e x ' y t )
a
cnseuahve v e c t r t
e e
l
d
)
FF
rerpretation o f the Divergence of a VectorField
fa
the idea of the divergence of a vector field
p o t h the divergencea n
gas. Here we focus on t
d curl are vector operators whose properties are
revealed by viewing a vector field as the flow of a fluid or
n of the curl on another page.
n read a similar discussio
he geometric properties of the divergence; you ca!
The divergence of a vector field is relatively easy to understand intuitively. Imagine that the vector field F pictured below givest h e
the fluid is exploding outward from the origin.
velocity of some f l u i d flow. I t appears that
?
.
?
.
>
Xr
~
.
- .
9
a t
4
?
?
y
?
é
¢
>
vif
4
<
¢
es
|
?
1°
2
=?
a
?we
e
e
=
t o n i
2
v
«
-
-
o
2
f e
S o w
1
*
-
ca
?
,
.
z i !
i
'
8
?
.
«
1
8
so
&
a
ao?
?;
°
,
r , t
?
.
This expansion o f fluid flowing with velocity field F is captured by the divergence o f F , which we denote d i v
FB.
The divergenceo f
the above vector field is positive since the flow is expanding.
In contrast, the below vector field represents f l u i d f l o w i n g so that i t compresses as j t moves t o w a r d t h e o r i g i n . Sincet h i s
o f this vector f i e l d is negative.
compression of fluid is the opposite of expansion, the divergence
a
?
t
i
t
a
F
?
a
0
?
.
ed
-
?
o
-
-
.
.
v i e
a
.
a
-
~
4
? t y
@
.
-?
:
e l e
?
«
e
*
?
?
a o s
1
a y s
a
.
.
~
.
x
.
2
The divergence is defined for both two-dimensional vector fields F ( z , y) and three-dimensional vector fields F ( z , y, 2). A threedimensional vector field F showing expansion of fluid flow is shown in the below applet. Again, because of the expansion, we can
conclude that div F ( x , y) > 0.
a
ngenceof
vector field simply measures how muuch the flow is expanding at a given point. It does not indicate in which
qr? ion the expansion is occuring. Hence (in contrast to the curl o f a vector field), the divergence is a scalar. Once you know the
emule for the divergence, it's quite simple to calculate the divergence of a vector
15.1.4
DEFINITION
I f F ( x , y , 2 = f . y, D i t 8 2 d
.
+ h(x, y, OK, then we de-
fine the divergence o f F, written div F, to be the function given by
divF= 50+ By *
&
_ Ff , 9g, Oh
d v F = V - +F5 )=+ 53,
Ex. 8
Find the divergence and the curl o f the vector field
F(x,
«4
=
= &
a i
2
(
=
x7 yl
+ 2y°2j + 32k
2»):
2 , 2 ?
=
dw F
y.2)
2
9
a
" f i s k
b k
x y,
oy?
r a ( a e ) s
( v y )
2
Jry,
5
,
@
x
|
aye
a
Z U ?
c a t y ,
2y
of h a
R S |
fye
xy dye Bz
>
= <
Se
i ( 0 - 4 )
P
37>
ay?
,
0,
-
-idxy
[ong
+
6y%2
&
+3
8,
[a
J R 2]
Ky 7 |
2 2TES
#@
-4 ( 0 - 9 )
x )
|
xy 2ye
+
k (o-**)
~
1,
7
divergence and curl of each vectorf i e l d .
pnd the
F(x, y,2) =
X74
~ 2] + yek
dw F = < 8 75 2 >
- a
Curl F =
e
e o
+
2
dx? p73y t ?
0
2,28
={fn
é o
dx
x>
Ax+O+y
2
=
|axty|
Cx p A Ve?
x
y a pg a ) - a e jl ae a / o l ta
&
3g
2 (ye)
+
5 *
-2 y?
a
(z-9)
t
=
yf
x?
YE
-J ( 0 - 0 )
§¢
aye
+
k (0-0)
re
a a e y |
j
f TET
F(x,yz)
b.
= az'it 2y*x" ft 527yk
B
d w F
=
{
t
,
e
Cart,
2
0
5
4
\
B a t )
+
B ( o e
[23 ,
hye"
(a,
a ,
C R ?
*
~
L o d
2
2
o x y
ne?
C52*, 3 0 2 % G u y
Carl =
>
OF
7
wo2y
us?
2?)
.
+
2
2
Set
( 8 4 )
|
.
Curl F
aye?,
S t y
a
:
)
L123, ayx®, S t y
L
.
D
i
a
2
=z
~=il
2
[ayiz? Sevy
-J
Sy
ax
x e
a
oT J - «
Bey
a2
ex
xe®
B y e Se7y
=
i
( g 7 - 0 )
= / @ ° ,
3 x
~
2 ,
j
(0~
axe")
A r y "
> |
+
b
xy?
0
)
£
oy,
y x
4
>