math 3 tutorial 7-12 problems.pdf

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MP2006/AE2002 Mathematics 3 Tutorials 7-12 Problems

Tutorial 7 (Week 8)

Problems 1 to 5 will be discussed during tutorial. Problems 6 to 9 areadditional problems for the students to attempt themselves. Guide forsolving those additional problems will be posted in the course website.Students who encounter difficulties in doing the additional problemsmay consult their tutors individually.

1. Show that (uv)w = (uw)v(vw)u (Hint. Let u= i+j+kv= i + j + k and w= i + j + k and work out separately the left

and the right hand sides of the given equation in terms of and Show that the two sides give the same expression.)

2. (a) With reference to a Cartesian coordinate system 0 find theplane that contains the points (2 1 3) (3 0 2) and (1 1 4) (Hint.One way of doing this is to find two vectors that lie on the plane anduse them to obtain a vector perpendicular to the plane.)

(b) A straight line perpendicular to the plane in part (a) passes throughthe point (2 52) Give a parametric representation for the straightline.

(c) Find the point where the plane in part (a) and the straight line inpart (b) intersect.

3. Find all unit magnitude vectors that are perpendicular to the vectorik. (Hint. Let i+j+k be a general vector that is perpendicular toik. Form a linear algebraic equation in and Find all solutions ofthe single linear algebraic equation. How can you ensure that i+j+kis of unit magnitude?)

4. Find a parametric representation for each of the following surfaces:

(a) 36( 1)2 + 252 + 9( + 2)2 = 36 (ellipsoidal surface)(b) 1 + 2 + 2 = 0 for 2 9

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5. Ifu= 1i + 2j + 3k, v= 1i + 2j + 3k and w= 1i + 2j + 3k,

show that

u (vw) = det

1 2 31 2 31 2 3

(Note. You should know how to calculate the determinant of a 3 3matrix from linear algebra. If not, you may like to take a look at Section7.7 of Kreyszigs Advanced Engineering Mathematics 9th edition.)

6. Let a, b, c and d be vectors in three-dimensional space. Prove ordisprove

(a b) (c d) = (a c)(b d) (a d)(b c)

7. With reference to a Cartesian coordinate system 0 sketch the sur-face =

p2 + 2 Find a parametric representation for the surface.

8. Find all points of intersection, if any, between the straight line givenby = 1 + 2 = 1 and = 3 + ( ) and the surface = 1 + 2 + 2

9. Find a parametric representation to describe the curve formed by the

intersection of the surfaces 2

+ 2

= 0 and 2 = 0

Tutorial 8 (Week 9)


1. For the vector function G() = 3 cos(2)i + 3 sin(2)j + k find:

(a) G G (b)G

G

(c) G

G

(d) |G

G

|

(Hint. The answer for part (d) can be deduced from parts (a), (b) and(c). What does your answer in part (c) tell you?)

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2. A point mass is moving on a plane. With reference to a Cartesian co-

ordinate frame 0 on the plane, the position is given by = 1 + and = 9 2 (m), where 0 is time in second (s).(a) Sketch the path travelled by the point mass during the period from = 0 to = 3 (s). (Hint. To get a clearer idea of the path of the pointmass, find a relationship between the and the co-ordinates of thepoint mass by eliminating )

(b) Calculate the velocity of the point mass at time = 5 (s).

(c) If the point mass represents a body of constant mass 5 kg, find the

total force acting on it.

3. Find a function ( ) such that

= (2 sin(2) + 4 cos(2) + 43 2)i+(2 sin(2) + 4 cos(2) + 54 12)j

4. If() = 2 + 2 + 32 + + 3 + 3 + + + 2 + 10 find Find all points () such that = 0. (Here 0 = 0i + 0j + 0k)Give a geometrical interpretation of all those points.

5. Find the function ( ) such that

= (32 + 4)i + (4 + 4 + 3)j and (1 3) = 5

6. Find a function () such that

= (43 + 4 + 8)i + (6 + 4 + 42 + 52)j+(8 + 4 + 10)k

(Note. When you integrate partially with respect to in find

you are holding and constants and the resulting constant ofintegration should be regarded as a function of and )

7. If and are partially differentiable functions of and show that

grad( ) = grad() + grad()

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8. The function () = 2 ++2 is such that

= 3i+2j

4k

at the point () = (1 1 2) find the constants and

Tutorial 9 (Week 10)


1. The air pressure in a particular room varies from point to point andis given by () = 32 + 22 exp(2)where () describes ageneral point (with reference to a Cartesian coordinate system).

(a) At the point (1 0 2) calculate the rate of change of pressure perunit distance in the direction of the vector i2j + 2k(b) At (1 0 2) find all directions along which the rate of change ofpressure per unit distance is zero. (Give the directions using vectors inthe form i + j + k)

(c) At (1 0 2) find the direction along which the rate of increase in the

pressure per unit distance is the greatest. What is the greatest rate ofincrease in the pressure per unit distance at (1 0 2)?

2. Find the straight line which is normal to the surface = 1 + 22 + 32

at the point (1 2 15)(Hint. First find a vector which is parallel to thestraight line.)

3. IfF and G are vector functions of and such that their componentshave first order partial derivatives, prove or disprove that

div(F G) = G curl(F) F curl(G).

4. If the velocity field of a fluid flow is given by q= , where is ascalar function of and , show that the flow is irrotational, that is,show that the fluid particles have zero rotational velocity. (Note. Aspointed out earlier on, q gives the rotational velocity of the fluidparticles. You may assume that the function has second order partialderivatives.)

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5. Find value(s) of the constant such that () = (2 + 2 + 2)

satisfies 2

= 0 at all points () except possibly at (0 0 0)

6. IfF and G are vector functions of and such that their componentshave first order partial derivatives, prove or disprove that

(F) G

= (11

+ 22

+ 33

)i

+(11

+ 22

+ 33

)j

+(11 + 2

2 + 3

3 )kF( G)

(ifF = [1 2 3] and G = [1 2 3])

(Note. F is a vector differential operator.)

7. The vector functions F( ) and G( ) are related to the scalar func-tion ( ) by

F( ) =

i

k and G( ) = j

Assume that the first and second order partial derivatives of exist.(Note that all the functions involved are independent of .)

(a) Evaluate div(F).

(b) If curl(F) = G show that div(grad()) = 8. Let and be scalar functions of the Cartesian coordinates and

Assuming that and are partially differentiable (at least twice)with respect to the Cartesian coordinates, prove or disprove:

curl( grad()) = grad() grad ()

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Problems 1 to 5 will be discussed during tutorial.

1. Calculate the line integralR

() for each of the following cases:

(a) () =

+ is the curve = 2 = 3, = for1 2 [Ans: 1

3(13

26 7

14) ' 13 365]

(b) () = is given by 2 + 2 = 1 0 0 = 0[Ans: 12]

(c) () = 2 + is the straight line segment between (1 2 3)and (5 0 2) [Ans: 13

21]

2. Find the length of the curve = 32, 0 1 = 0 [Ans:13

27

13 8

27' 1 44]

3. Calculate the line integralR

F r for each of the following cases:

(a) F = 2i j + k is the straight line segment from (0 1 2) to(

1 0 5) [Ans: 1]

(b) F = i 2j + k is given by = sin() = 2 cos() = from = 0 to = 4 [Ans: 9

4+ 1

322 ' 2 56]

4. Use Greens theorem to find the work done by the two-dimensionalforce field F = (32 + 2)i+ (3)j along the curve if is one fullround of a circle in the clockwise direction and the radius of the circleis 2 units. [Ans: +12]

5. Verify that Greens theorem is true for F = (i +j) and the region bounded by the curve as sketched in Figure 27.

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Figure 27


Problems 1 to 4 will be discussed during tutorial.

1. Check that the two dimensional force field F = 2( 1)i + (2 + 2)jis conservative by partially differentiating the components of F in asuitable manner. Find a potential function for F and calculate thework done by F on a body which moves from the point (2 0) to (1 1)on the plane. [Ans: work done = 5]

2. Sketch the surface given by = 2+2 1 4 Evaluate a suitablesurface integral to compute the area of [Ans: 17

6

17 56

5 '30 85]

3. Calculate the surface integralRR

() for each of the following

cases. (You must be able to visualise the surface in order to do theintegration properly.)

(a) () = is the portion of the plane + 4 + = 8 insidethe cylindrical region 2 + 2 = 1 [Ans: 32]

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(b) () = is the surface 2 + 2 = 1 0

12

0

0 4 (Hint. Rewrite 2 + 2 = 1 as = +1 2 since 0 Project the surface along the axis perpendicularly ontothe plane.) [Ans: 2]

4. The surface is given by 2 + 2 + 2 = 4 1 Find the unit normalvector n to such that the component ofn is negative. Computethe flux of the vector field u = i + j + k across in the direction

ofn. [Ans: 152

= 23 56]


Problems 1 to 4 will be discussed during tutorial. Problem 5 is optional(to be attempted by only the highly motivated students).

1. For each of the following cases, use Gauss theorem to compute the fluxof the vector field u across the surface (bounding the region ) inthe direction of the outward unit normal vector to

(a) u = (2 + )i + j+ k is defined by 0 1 1 11 2 [Ans: 11

2]

(b) u = 2i + (3 + )k is given by 2 + 2 9 0 5 [Ans:225]

2. Verify that Gauss theorem is true for u = i+ j+k in the boundedregion defined by 2 + 2 + 2 4 0

3. Verify that Stokes theorem is true for F = i+ j+k on the surface which is given by the portion of the plane + 3 + 2 = 6 in thefirst octant. (Choose a unit normal vector to the surface and thenassign the correct direction to the curve which is along the edge ofthe surface .)

4. Find the constant such that the force field F() = (2 + )i +j + k such that F is conservative. Hence find a potential functionfor F and compute the work done by F along any path with (2 1 2)and (3 0 6) as starting and ending points respectively. [Ans: = 12work done = 13]

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5. Show that Greens theorem as given by

I

[( ) + ( )] =

ZZ

can be rewritten asI

[( )( )] [1 2] =ZZ

[( )( )]

where [1 2] is the unit normal vector to pointing away from the

region (Can you see that the rewritten Greens theorem is really thedivergence theorem in two-dimensional form?)

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math 3 tutorial 7-12 problems.pdf

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