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Tutorial 1

Question 1Calculate the determinant of the five matrices and state which are singular.

A=

2 0 11 2 2

3 2 4

, B=

2 1 2

1 1 54 2 3

, AB2, A + B, AB + A2.

Question 2

(a) Let A =

0 0 1

0 0

0 0

. Verify thatA3 = I3, where I3 is the identitymatrix of order 3. Deduce that A is non-singular and hence, compute A1.

(b) Find the value(s) ofxfor which the following matrix is singular:

x+ 3 2 1

x 3 x 4

2 x 1 2 1

.

(c) Given thatM =

2 + 3 1

2 + 3 1

2 + 3 1

, use properties of determinant to show

that|M| = ( ) (+) ( ).Question 3An n n matrixA= (aij) is called orthogonal ifAAT = I.

i Show that the following matrix is orthogonal:

12

16

13

12

16

13

0 2

6

13

ii Show that ifA is orthogonal then det(A) =1.

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AnswersQuestion 1

|A| = 12,|B| =3,|AB2

| = 108|A + B| = 0 (singular),|AB + A2

| = 0(singular).Question 2

(a)

0 1 0

0 0 1

1 0 0

.

(b)8

3, 4.

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Tutorial 2

Question 1Let be a complex cube root of 1 (this means 3 = 1) with = 1. Prove that1 ++2 = 0.Letting A be the matrix

1 1 11 2

1 2

Determine A2 and A1.

Question 2

For each v , define the matrix A(v) by

A(v) =

1 0 0v 1 0

12

v2 v 1

Show that for all v, w we have A(v+ w) = A(v)A(w). Deduce that eachmatrix A(v) is invertible.

Question 3Determine how the rank of the real matrix

3 1 21 2 11 0 12 b 1

depends on the real number b.

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Question 4Find the rank of the following matrices:

(a)

1 1 11 1 2 +

1 2

, where 1 ++2 = 0.and 3 = 1.

(b)

2 1 2 1

2 2 4 4

0 1 2 3

4 5 1 0 1 1

, (c)

1 1 1 2

1 3 1 4

1 5 1 7

.

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AnswersQuestion 1

A2 =

3 0 00 0 3

0 3 0

, A1 =

1/3 1/3 1/31/3 2/3 /3

1/3 /3 2/3

Question 3(i) rank= 2 f o rb = 3 and rank= 3 f o rb= 3Question 4(a) 2, (b) 2, (c) 3.

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Tutorial 3

Question 1Solve the following system of equations using Gaussian elimination:

(a)x+ 2y 4z= 4

2x+ 5y 9z= 103x 2y+ 3z= 11

(b)x+ 2y 3z= 1

3x+y 2z= 75x+ 3y 4z= 2

(c)x+ 2y 3z= 1

2x+ 5y 8z= 43x+ 8y 13z= 7

Question 2

Consider the system2x+ 2y+ 3z= 0

x+y+ 3z= 0

x+ 2y+ 2z= 0.

Find sufficient conditions on for the system to have

(i) unique solution,

(ii) infinite number of solutions.

Question 3Find the inverses of the following matrices using elementary row operations:

A=

1 1 01 0 1

1 0 0

, B=

1 1 22 1 1

2 2 1

, C=

1 1 10 1 2

3 1 1

.

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Question 4Let A be each of the following matrices in turn:

2 1 20 0 1

0 1 0

,

1 1 01 3 0

1 4 1

,

2 1 11 2 1

1 1 2

.

(a) Find all the eigenvectors ofA; determine whether A is diagonalisable and,if so, find an invertible real matrix Xfor which X1AXis diagonal.

(b) In the case of the first matrix,

(i) find the eigenvalues and eigenvectors ofA5,A + 7Iand (A - 3I)1.

(ii) Show that A1 = A 12A2 +12 Iand hence, or otherwise, determineA1

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AnswersQuestion 1

(a) x = 2, y = 1, z= 1, (b)inconsistent (c)x= 3 t, y= 2 + 2t, z=t.Question 2(i) = 4 (ii)= 4.Question 3

(a)

0 1 00 0 11 1 1

, (b)

1/10 3/5 3/101/5 1/5 2/53/10 1/5 1/10

, (c)

1/3 0 2/31 1 0

1/3 1 1/3

.

Question 4

(a){2, (1, 0, 0)t

} , {1, (3, 1, 1)t

} , {1, (1, 3, 3)t

} and X =

1 1 3

3 0 13 0 1

{1, [0, 0, 1]} , {2, [1, 1, 1], [1, 1, 1]}

{1, [1, 1, 0], [0, 1, 1]} , {4, [1, 1, 1]} and X=

1 1 0

1 1 11 0 1

.

(b)(i)Each eigenvalue is raised to power 5 and the corresponding eigenvectorsare unchanged; add7 to each eigenvalue and corresponding eigenvectors are un-changed; subtract 3 from each eigenvalue and take reciprocal of each resulting

value and corresponding eigenvectors are unchanged.

(ii) A1 =

1/2 1 1/20 0 1

0 1 0

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Tutorial 4

Question 1Test the following series for convergence:

(i)n=1

en

3n+2,

(ii)n=1

n7n2

,

(iii)

n=1

2(2n

1)(2n+1)

,

(iv)n=1

nn3+1

,

(v)n=1

1n2+1

n21

(vi)n=1

sin1n

,

(vii)n=1

nen,

(viii)n=1

n!2n1!

(ix)n=1

n!1.5.9...(4n3)x

n,x >0.

Question 2Given that y = (5 2n

)n, n > 0, by applying LHospitals Rule, show that

limn

ln y = 25

. Hence, or otherwise, determine whether the seriesn=1

5n2n

nconverges or diverges.

Question 3Use the Taylor series to find a quadratic approximation to each of the followingfunctions at the specified points:

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(i) sin(3x+ 2y) at (6

, 0),

(ii) cosh x cosh y at the origin.

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Answers1.(i)convergent; (ii)divergent; (iii)convergent;

(iv)convergent; (v)divergent; (vi)divergent;

(vii)convergent; (viii)convergent; (ix)convergent for 0 < x

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Tutorial 5

Question 1Thetriple vector product(AB)C) andA (BC) are usually not equal,although the formulae for evaluating them from components are similar :

(A B) C = (A C)B (B C)A. (1)

A (B C) = (A C)B (A B)C. (2)Verify each of the formula for the following vectors by evaluating its two sidesand comparing the results.

A B Ca) (1, 1, 2)t (1, 0, 1)t (2, 4, 2)tb) (1, 1, 1)t (2, 1, 2)t (1, 2, 1)tc) (2, 1, 0)t (2, 1, 1)t (1, 0, 2)t

Question 2Show that ifA,B,Cand D are any vectors, then

(a)

A (B C) + B (C A) + C (A B) = 0.

(b)

A B = (A B i)i + (A B j)j + (A B k)k.

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Question 3

(a) Given that A = 2i + 2j k, B = i + k and C = i + j + k, find thescalar and vector projection ofB C in the direction ofA.

(b) Show that if the lines

x = a1s + b1, y = a2s + b2, z = a3s + b3, < s

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AnswersQuestion 1

(a) (10, 0, 10)t

(1) (12, 4, 8)t

(2) (b) (10, 2, 6)t

(1) (9, 2, 6)t

(2)(c)(4, 6, 2)t (1) (1, 2, 4)t (2)Question 3

(a)23

3,

2

3(1, 1, 1)t

Question 4

(b)1

6 (4, 5, 1)t = 1

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AnswersQuestion 1

b(i) (x, y, 0)t

, 0, (ii)(x, 2y, z)t

, 0.Question 2

(i) (2z 3y, 3x 1z, 1y 2x)t, (iv)2

Question 3(i)52, (ii)192, (b)2193 in the direction = 30015Question 416

49 24

49

3

Question 5

(i)

2

4

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Tutorial 7

Question 1

(a) Sketch the region of integration of

10

2xx

dydx.

(b) Exchange the order of integration to express the integral in part (a) in termsof integration in the order dxdy and evaluate it in that case.

(c) Find

y

x2 +y2dxdy, where is the shaded region shown in Fig.(1).

0.5

1

1.5

2

2.5

3

3 2 1 1 2 3

x

Figure 1: Region

(d) Find

ydydx, where is the region bounded by the triangle with ver-

tices at (2, 0), (0, 1) and (2, 0)Question 2

Sketch the corresponding region of integration and evaluate by changing order ofthe integration :

(a)

30

9x2

dydx,

(b)

10

x0

2xy

1 y4dydx.

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Question 3

Find 1

x

ay

bdydx, where is in the first quadrant bounded by the

axes and the line x

a+

y

b = 1.

Question 4Consider the following integral

I =

20

y0

cos

(x2 +y2)

8

dxdy +

22

4y20

cos

(x2 +y2)

8

dxdy

(i) Sketch the region of integration defined by the sum of these two integrals.

(ii) By reversing the order of integration, rewrite Ias one double integral.

(iii) By using polar coordinates, show thatI = 1.

AnswersQuestion 1

(b)

10

yy/2

dydx +

21

1y/2

dydx, 1

2, (c) 4, (d)

2

3.

Question 2

(a) 18, (b) 1

2.

Question 3

ab6

.

Question 4

I =20

4x2x

cos

(x2 +y2)

8

dy dx.

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Tutorial 8

Question 1Consider the integral

I =

10

10

1

(1 +y2x)(1 +x)dxdy.

The transformationx= u2andy=

v

u,

is applied to I.Show that the limits in the u v plane are : u = 0, u = 1 and v = 0, v = 1.sketch the region in the u

v plane and hence evaluate the above integral by

integrating over an appropriate region in the u v plane.

Question 2

Consider the integral I =

R

(x+y)(x2 +y2)

x4 dA, where R is the region

bounded by the lines y = 0, y = x and x + y = with > 0 . By ap-

plying the transformation u= x + y and v =y

x, sketch the corresponding region

of integration in the u v. Hence, evaluate I.

Question 3

Show by means of a diagram the area over which the double integral is taken,

I =20

2yy2

x+y

(x+ 1)2e2(x+y)dxdy. Apply the transformation of variableu = x +y

and v = y 1x+ 1

to this integral and sketch the region of integration in the u vplane. Hence, evaluate I.

Question 4If f(x, y) can be written as f(x, y) = F(x)G(y), then the integral of f over arectangle R: a x b,c y d can be evaluated as a product by the formula

R

f(x, y) dxdy=

b

a

F(x)dx

d

c

G(y)dy. (3)

Use Eq.(3) to evaluate the following:

(i)

10

2

0

6x(cos(y))2dydx.

(ii)

21

10

x3y3/2dxdy.

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Question 5

Let I=

0

e(x2)dxand I2 =

0

0

e(x2+y2)dxdy.

Evaluate the last integral using polar coordinates and solve the resulting equation

to show that0

ex2

dx=

2 .

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AnswersQuestion 1

2

16Question 24

3Question 31

4

e4 1

Question 4

(b)(i) 3

4,(ii)

2 24

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Tutorial 9

Question 1Evaluate the following triple integrals(a)

10

1x0

2x0

xyzdzdydx.

(b)

2

0

10

20

r2zsin dzdrd.

(c)

0

4

0

sec0

sin2ddd.

Question 2

Using triple integrals, find the volume of the tetrahedron bounded by the coor-dinate planes and the plane x+y+z= 1.

Question 3Calculate the volume of the region bounded by the following surfaces:

(i) z= 0, x2 +y2 = 2 and x+y+z= 3.

(ii) The cylinderx2 +y2 = 16 and the planes z= 0 and z+y = 4.

(iii) The spherex2 +y2 +z2 = 16 and the cylinder x2 +y2 = 9.

Question 4Find the area of that portion of the surface of the cylinder x2 + y2 = 9 which liesin the first octant between the planes z= 0 and z= 2x.

Question 5Find the surface area of the part of the sphere x2 + y2 + z2 = 4 inside the upperpart of the cone x2 +y2 =z2.

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AnswersQuestion 1

(a) 13240

(b) 2

3(c) (2 2)Question 21

6Question 3

(i)3

2

43

2, (ii)64, (iii)

2563 32 3

Question 4

18Question 54

2 2

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Tutorial 10

Question 1

For F = x3yi +y2j, findC

F dr along the curve y= x2 from (0, 0) to (1, 1).Question 2

Evaluate

C

(xy)dx+ (yz)dy over the line segment C from P(1, 1, 1) toQ(2, 4, 8).Hint:C is defined as follows :

PQ= (1, 3, 7)t and any point (x,y,z)along Cis thus definedas

(x= 1 +t, y= 1 + 3t, z= 1 + 7t, where0

t

1.

Question 3

(a) Show thatF= (3x2 6y2)i + (12xy+ 4y)j is conservative.(b) Find such F = ( is said to be a potential function for F).(c) Let Cbe the curve x = 1 + y3(1 y)3,0 y 1. Calculate

CF dr.

Question 4Let F= (ax2y + y3 + 1)i + (2x3 + by2x + 2)jbe a vector field, where a and b are

constants.(a) Find the values ofaand bfor whichF is conservative.

(b) Use the values ofaand b from (a) to find f(x, y) such that F =f.(c) Using the values ofa and b from part (a), compute

C

F dralong the curveCsuch that x= et cos t,y = et sin t, 0 t .

Question 5

(a) Write down the contour integral ofF= (5x+ 3y)i + (1 + cos y)j, counter-

clockwise around the unit circle centered at the origin, in the form ba

f(t)dt.

(Do not simplify the integral.)

(b) Evaluate the line integral using Greens theorem.

Question 6Consider the region Renclosed by the xaxis,x= 1 and y = x3.Travelling in a counterclockwise direction along the boundary CorR, callC1 theportion ofCthat goes from (0, 0) to (1, 0), C2 the portion ofCthat goes from(1, 0) to (1, 1) and C3 the portion ofCthat goes from (1, 1) to (0, 0).

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(a) Using Greens theorem, find the total work ofF = (1 +y2)i around theboundaryCofR, in a counterclockwise direction.

(b) Calculate the work ofF along C1 and C2.

(c) Use parts (a) and (b) to find the work along the third side C3.

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AnswersQuestion 1

1/2Question 213Question 3(b)f=x3 6y2x+ 2y2(+constant), (c) -4Question 4(a) a = 6, b= 3, (b) f= 2x3y+y3x+x+ 2y(+constant), (c)e 1.Question 5 20

(5cos t + 3sin t)( sin t)dt + (1 + cos(sin t))cos tdt, (b)3.Question 6(a)1/7, (b)alongC1,it is 1, and along C2,it is 0(c)8/7.

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Tutorial 11 - Divergence theorem

Question 1

(a) if

is any closed surface enclosing a volume V and = (xi + yj + zk), usethe Gauss-divergence theorem to prove that

ndS = 3V.

(b) Given thatr= xi +yj + (1 z)k, evaluate

r ndS

over the whole boundary of the region bounded by the paraboloid z =13 x2 y2 and the plane z= 9.

Question 2Calculate the flux ofF = xi + yj + (1 + 3z)k out of the portion of the spherez2 = 4 x2 y2 in the first octant in the direction away from the origin

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Question 3

Figure 2: Plot showing the paraboloid z = 1 x2 y2 and z = 0.

Let S be the curved part of the surface formed by the paraboloid z =1 x2 y2 lying above the xy plane as shown in fig.(2). The basis of the solidformed is the unit disc in the xyplane. Further, let F = xi + yj + (1 2z)k.

(i) Using the divergence theorem, find the flux ofFout of the solid.

(ii) Using direct calculation and taking the upward direction as the one forwhich the flux is positive, compute the flux ofF through the unit disc inthe xy plane (n =k).

(iii) Hence, calculate the flux ofFthrough S.

Question 4Use the divergence theorem to compute the flux of F = (1+ y2)j out of thecurved part of the half-cylinder bounded by x2 + y2 = a2, (y 0), z = 0,z = b and y = 0.

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AnswersQuestion 1

8Question 28

3Question 3(i) 0, (ii)2, (iii) 2.Question 4

2ab + 4a3b

3 .

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Tutorial 12 - Stokes theorem

Question 1Let F = (6y2 + 6y)i + (x2 3z2)j x2k.Calculate Fand use Stokestheorem to show that the work done by Falonga simple closed curve contained in the plane x + 2y + z= 1 is equal to zero.

Question 2Let F= 2xzi +y2k.

(a) Calculate curlF.

(b) Show that the RcurlF ndS = 0 for any finite portion R of the unitsphere x

2

+y2

+z2

= 1.(Take the normal vector npointing outward).

(c) Show thatC

F dr = 0 for any simple closed curve Con the unit spherex2 +y2 +z2 = 1.

Question 3Let F = (xz, yz + x, xy)t.

(a) Find F.(b) Let C be the simple closed curve, oriented counterclockwise when viewed

from above,x

y + 2z= 10. The projection ofCon thexy

plane is the

circle (x 1)2 + y2 = 1. Use Stokes theorem to compute CF dr.Question 4Use Stokes theorem to find

C

Fdr, whereC = C1

C2,C1is the circumferenceof the semi circle of radius a, above the xaxis andC2 is the line segment [a, a]on the xaxis and F = y2(a2 z2)i + ax2(a 3z)j + x2y2k.

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AnswersQuestion 2

F = (2y,2x, 0)t

. Question 3(a) F = (x y,x y, 1)t.(b) .Question 44a5

3 .

END-OF-TUTORIAL

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