vector

398 Vectors
5. Find the directional derivative of the scalar function of (x, y, z) = xyz in the direction of the outer
normal to the surface z = xy at the point (3, 1, 3).
Ans.
27
11
6. The temperature of the points in space is given by T(x, y, z) = x 2 + y 2 – z. A mosquito located at
(1, 1, 2) desires to fly in such a direction that it will get warm as soon as possible. In what direction
should it move?
Ans.
1 (2 i
2 j
k )
3
7. If (x, y, z) = 3xz 2 y – y 3 z 2 , find grad at the point (1, –2, –1) Ans.
(16i 9j 4 k)
8. Find a unit vector normal to the surface x 2 y + 2xz = 4 at the point (2, –2, 3).
1
Ans. ( i 2 j 2 k
)
3
9. What is the greatest rate of increase of the function u = xyz 2 at the point (1, 0, 3)? Ans. 9
10. If is the acute angle between the surfaces xyz 2 = 3x + z 2 and 3x 2 – y 2 + 2z = 1 at the point
(1, –2, 1) show that cos = 3/7 6 .
11. Find the values of constants a, b, c so that the maximum value of the directional directive of
= axy 2 + byz + cz 2 x 3 at (1, 2, –1) has a maximum magnitude 64 in the direction parallel to the
axis of z. Ans. a = b, b = 24, c = –8
12. Find the values of and µ so that surfaces x 2 – µ y z = ( + 2)x and 4 x 2 y + z 3 = 4 intersect
orthogonally at the point (1, –1, 2). Ans. = 9 , 1
2
13. The position vector of a particle at time t is R = cos (t – 1) i + sinh (t – 1) j + at 2 k. If at t = 1,
the acceleration of the particle be perpendicular to its position vector, then a is equal to
1
(a) 0 (b) 1 (c)
2
5.29 DIVERGENCE OF A VECTOR FUNCTION
The divergence of a vector point function F
(d)
1
2
(AMIETE, Dec. 2009) Ans. (d)
is denoted by div F and is defined as below.
Let F = Fi 1 F 2 j Fk 3
div F1 F2 F F = . F i j k ( iF1 jF2 kF3)
= 3
x y z
x y z
It is evident that div F is scalar function.
5.30 PHYSICAL INTERPRETATION OF DIVERGENCE
Let us consider the case of a fluid flow. Consider a small rectangular parallelopiped of
dimensions dx, dy, dz parallel to x,y and z axes respectively.
Let V Vx i Vy j Vz
k
be the velocity of the
fluid at P(x, y, z).
Mass of fluid flowing in through the face ABCD in unit time
= Velocity × Area of the face = V x
(dy dz )
Mass of fluid flowing out across the face PQRS per unit time
= V x
(x + dx) (dy dz)
Vx
= Vx
dx ( dydz)
x
Y
Net decrease in mass of fluid in the parallelopiped
corresponding to the flow along x-axis per unit time
O
Z
A
C
D
dz
V x B
S
P
R
Q
X