Vector Calculus Practice Questions and Answers

ENGINEERING MATHEMATICS



VECTOR CALCULUS




QUESTIONS AND ANSWERS



ZFS Co.

,I .
(a) i = pa =
-
54 + y

i = j2y) -
Sy + j) =
-ji j +




xe
-
24
f(x , y) =




23
fuln y) ,
= 2xe fy(m y)
,
= -
2n2eY
,




Dif(n y) ,
=
fu(n y)u , ,
+
fy(n y)u ,




= zne (52) -
zurez
(s2
ja
-
20
-




Duf (2 0) ,
=
126


-
=
#




Unit vector
Max . wate of
change = 10 f(x, y)/

=
( fu(u , y)i +
fy(u y(j)
,




=
Ifu (2 01i ,
+
fy (2 015/
,




= (4i -
8j/

=
xs(x(i 25) -




Is
=
(4 -

25)#




Max
. rate of change
=




14i -
8j) = 180

= 4)s#

,(b) i = p = Y -



T
i = (1/) + ( -



5)
=
(1/(5)(ii 55) -




=
(1/()(ii 15) -




= (i (j) -




=
ji jj -




f(u , y) = sin(2x y) -




fu (u y) ,
= 2 cos(2n -
y)



fy(n , y) = -
cos(2x -

y)




Daf(n y) ,
=
fu(n y)u
, , + fy(n y)4 ,




Put ( j 5)
-

,
= -

j(j) + 2(ss)
2/y
-
-




=

Is
-
5/3
I
215

-
=




Unit Vector
max .
rate of
change = 13 f(m y)l ,




=
(fn(x y)i ,
+
fy(x y)j) ,




=
Ifu)- (j)i , + ty) 5 5)5)
-


,




= 1 -
63i + 15
ish i 125)
= +




=
(33jsh -i 1425) +




=
js)-2i +
j)#


Max
. rate of
change
=




1 -
63i + (j) =

, ·

old
·

.
2 (a) h(n , y) = 300 -
2n2 -

3y

hx(n y) = 42
,




hy (n, y) =
by




34(x , y) =
hu(4 9)i
,
+
my(4 , 9)]

-
=
16 -

545

-
=
(16 +
54])

= 16i +
545#



(b) Perpendicular ,
and change sign for either term



v =
2(27i -

85)

. v = 27i -
85#