Page 78 - Engineering Mathematics Workbook_Final

Vector Calculus

(a) 0 (b) 2 161. If f = 2x + 3y + 4z , the value of
2
3

)
(c) 4 (d) 6 line integral (grad f . Dr evaluated
C
[GATE-2016-EE-SET 2] over contour C formed by the segments

159. A scalar potential  has one following (-3, -3, 2) → (2, -3, 2) → (2, 6, 2) →
$
$
$
gradient;  = yz i + xz j + xyk . (2, 6, -1) is _______. [GATE-19-EE]

c    
Consider the integral dr on 162. Let a and a be unit vectors along x
y
x
$
$
$
the curve r = x i + y j + zk . The curve and y directions, respectively. A vector
−
C is parameterized as follows: function is given by F = a y a x .
y
x
 x t The line integral of the above function
=
  F  dl along the curve C, which
 

=
3
 y t 2 and 1 t 
 C
 z = 3t 2 follows the parabola y = x as shown

2

below is __________ (rounded off to 2
The value of the integral is _________
decimal places).
[GATE-2016]
160. As shown in the figure, C is the arc from
the point (3, 0) to the point (0, 3) on the
circle x + 2 y = 2 9. The value of the
integral
)
)
(  y + 2yx dx + ( 2xy x dy is
2
+
2
c
[GATE-20-EE]
_____________
163. Consider the line integral
(up to 2 decimal places).
 (xdy − y dx ) the integral being
C
taken in a counterclockwise direction
over the closed curve C that forms the
boundary of the region R shown in the
figure below. The region R is the area
enclosed by the union of a 2 3
rectangle and a semi-circle of radius 1.

The line integral evaluates to
[GATE-18-EE]

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