## Transcribed Text

1. Surface integral
(a) [10 marks] Evaluate the surface integral
//s
y2 dS. S is the part of the plane . + y + 2 = 1
that lies in the first octant.
(b) [10 marks] Evaluate the surface integral
F - ds for the given vector field F = -
S
+ zk. S is the part of the plane . + y + 2 = 1 that lies in the first octant.
2. Green's theorem
(a)
[10
marks]
Use
Green's
Theorem
to
evaluate
the
line
integral
along
the
given
positively
oriented curve.
ey dar + 2.re³ dy
C is the square with sides x = 0, . = 1. y = 0 and y = 1.
(b)
[10
marks]
Use
Green's
Theorem
to
find
the
work
done
by
the
force
F(r.y)
=
in moving a particle from the origin along the to (1.0), then along the line segment
to (0,1), then back to the origin along the y-axis.
3. Curl and divergence
(a) [10 marks) Find the curl and the divergence of the vector fields F(x.y.z) =rydi+ny?:j+ =
and F(x.y.z) = e singi + cos yj + zk.
(b) [10 marks] Determine whether or not the vector field F = e2i + it .re²k is conservative. If
it is, find a function f such that F = Vf.
4. Vector field integrals
(a) [10 marks] Use Stokes' Theorem to evaluate //s curlF - dS where F(x.y,2) = +
sin ry j + ryck. and S is the part of the cone y2 = >2 + 22 that lies between the planes
y I 0 and y = 3. oriented in the direction of the positive y-axis.
(b) [10 marks] Use the divergence theorem to calculate the surface integral
F - dS, that
S
is, calculate the flux of F = xyi - . !] j - x2 yzk across S. S is the surface of the solid
bounded by the hyperboloid 22 + y - z2 - 1 and the planes 2 = 2 and 2 - 2.
5. Laplace transform
(a) [10 marks] Find the Laplace transform L(sintcost + (2t).
(b) [10 marks] Find the inverse Laplace transform + 2s
s2
+
2
- s2 + 4s - 4

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