ADDIS ABABA UNIVERSITY

Department of Mathematics
Math2042 Applied Mathematics III
Worksheet 3

1. Find lim f (t ) if it exists.

t 0

sin t t2 1  cos t t
a) f (t )  i e j  t k
2t
b) f (t )  t 2 i  j k
2t e  3t t 1 
        
2. Let u (t )  3t i  t k , v (t )  t 2 i  2 j  (t 1) k and w(t )  3t 2 i  t 3 j  t k . Find
      
a) ( u  v ) b) ( u  v ) c) [ v  (u  w)]
3. Find a parametric representation of
a) the curve (x  2) 2  y 2  9, z  - 1 in counterclockwise direction.
b) the ellipse 4x 2  y 2  4 in the xy -plane.
4. Find the length of the curve parametrized by
   
a) r (t )  e t cos t i  e t sin t j  5k for 0  t  1 .
   
b) r (t )  e t i  e t j  2t k for 0  t  1
     
5. Find the definite integrals a)  r (t )dt for r (t )  sin t i  cos t j  t k
0
1    
b)  r (t ) dt for r (t )  t i  1  t j  e t k
0
 
6. Find the unit tangent vector T , the principal normal vector N , the binormal vector, the
curvature and radius of curvature to the curve traced out by
   
a) r (t )  e t i  e t sin t j  e t cos t k at t = 0
  1     1 
b) r (t )  t 2 i  ( t 2 1) j  t k at t = 0 c) r (t )  t i  e 2t j at t  ln 2
2 2
7. Prove that the following vector functions are solenoidal (or incompressible)
     
a) f ( x, y, z )  yz i  xz j  xy k b) g ( x, y, z )  3x 2 i  y 2 j  (2 yz  6 xz ) k
  
8. Find the divF , curl F and curl (curl F ) of the following vector fields
       
a) F ( x, y, y)  xe y i  2e yz j  xe y k b) F ( x, y, z)  xy 2 i  ( y  2 z) j  (3 zx)k

c) F ( x, y, z ) = grad(x2 + y2 + z2 – xyz)
 
9. Determine whether or not F is a gradient of some function f. If F =  f, find f.
      
a) F ( x, y)  y 3 i  3xy 2 j b) F ( x, y, z )  yz i  xz j  ( xy  1) k
   
c) F ( x, y, z)  yzexy i + xze xy j + (e xy + z 2 ) k
  
10. Show that curl ( fF ) = f curl F + f  F , where all the partial derivatives of the scalar
valued function f exist.
   
11. Show that F ( x, y, z )  y i  ( z cos yz  x) j  ( y cos yz ) k is irrotational and if it exists find

the potential function of F . 
  
12. Show that the curvature of the helix r (t )  a cos ti  a sin tj  btk where a  0, t   is a
constant.
 ADDIS ABABA UNIVERSITY
Department of Mathematics
Math2042 Applied Mathematics III
Worksheet 4
1. Evaluate the following line integrals

a) C
(2 xz  xy 2 1) ds , where C is the line segment from (-1,0,2) to (3,2,1)

b)  C
y 2 dx  3xdy , where C is the triangle with vertices (0,0), (1,0) and (1,1).

c) C
xydy  y 2 dx , where C is the square with vertices at (-1,0), (1,0), (-1,2) and (1,2).
     
2. Evaluate C
F  dr , where F ( x, y, z ) = xyi - x 2 j + sin zk , where C is the line segment
from P(1,3,1) to Q(3,1,2).
( 2,1, 2 )
3. Show that the line integral  (1, 0, 0 )
(2 x sin y  e 3 z )dx  x 2 cos ydy  (3xe 3 z  4)dz
is independent of path and then evaluate it.

4. Use Green’s Theorem to evaluate the line integrals.

a) C
xydx  ((1 / 2) x 2  xy )dy , where C is the upper half of the ellipse x2 + 4y2 = 1
and the x -axis on [-1,1] oriented positively.

 (e x  y 2 )dx  (e y  x 2 )dy, where C is the boundary of the region between y  x

2
b)
C

and y  x , oriented counterclockwise.

  
5. Verify Green’s Theorem for F ( x, y)  y i  3x j , where C is parameterized by
  
r (t )  2 cos t i  2 sin t j for 0  t  2 . Using Green’s Theorem, find the area of the
region enclosed by C.

6. Evaluate the surface integral  f ( x, y, z) dS , if

S

a) f ( x, y, z)  x  1 and S =  is the portion of the plane 2 x  y  3z  6 in the first

octant.
b) f ( x, y, z )  1  4 x 2  4 y 2 and S is the part of the paraboloid z  x 2  y 2 below the
plane z  y .

  
7. Find the flux of the vector field F across S oriented upward,  F  n dS
S
if
    
a) F ( x, y, z )  yi  xj  4k and S is the paraboloid z  9  x 2  y 2 above the
xy-plane.
   
b) F ( x, y, z )  i  j  2k and S is the lower hemisphere of x 2  y 2  z 2  1 .
  
8. Use Gauss Divergence Theorem to evaluate   n dS , where n is the unit normal vector
S
F

to S directed outward if,

   
a) F ( x, y, z )  x 2 i  xyj  2 xzk ; S is the tetrahedron with vertices (0,0,0), (1,0,0),
(0,1,0), (0,0,2).
   
b) F ( x, y, z)  xi  2 y 2 j  z 2 k ; S is the boundary of the solid bounded by the
cylinder x 2  y 2  4 between the planes z = 0 and z = 3.
   
c) F ( x, y, z)  (2 x  3z)i  ( xz  y) j  ( y 2  2 z)k and S is the surface of the sphere
having center at (3,-1,2) and radius 3 units.
     
     
2 xz
9. Evaluate F dS , where F ( x, y , z ) xy i ( y e ) j sin( xy ) k and S is the surface
S

of the region D bounded by the parabolic cylinder y  1  x 2 and the plane z  0 , y  0

and x  y  z  2 .
   
10. Verify Stokes’ Theorem for F ( x, y, z)  (2 x  3 y)i  y z 2 j  y 2 zk and S in the upper half
(above the xy-axis) of the unit sphere.

11. Evaluate the line integral  F  dr using Stokes’ Theorem, where
C
   
a) F ( x, y, z )  xyi  yzj  zxk ; C is the triangle in the plane x  y  z  1 with
vertices (1,0,0) , (0,1,0) , (0,0,1) oriented positively as viewed from above.
   
b) F ( x, y, z )  ( z  y)i  yj  xk ; S the upper part of the hemisphere of

x 2  y 2  z 2  4 and C is oriented by an upward normal n .
     
12. Use Stokes’ Theorem to evaluate 
S
( curl F )  n dS , where F ( x, y , z )  yi  zj  xk ,

where S is the surface z  1  x  y 2 for which z  0.

2

  
13. If F  f and  2 f = 0, show that for a closed surface S =  D,  )  n dS =  F
2
( f F dV
S D