Dougs Engineering Mathmatics Cousework for Semester 1 Final

8/2/2019 Dougs Engineering Mathmatics Cousework for Semester 1 Final

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Faculty of Engineering

Science and

Built Environment

Course: BEng Building Services Engineering

Mode: Part Time

Level: Two

Unit: Engineering Mathematical Methods Coursework

Unit Code: SCE-2-203

Date: Semester 1, 2010

Prepared By

Douglas Buchan


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Question 1

(a) Arrange the equations below into diagonally dominant form giving a

motivation for doing so. (5 Marks)

=

By arranging the matrix into a diagonally dominant form convergence is guaranteed,

when using both the Jacobi and the Gauss-Seidel iteration schemes, without

arranging the matrix into diagonally dominant form the equation may not converge.

=

(b) Perform one cycle of the Gauss-Seidel iteration scheme for solving a

system of linear equations on your diagonally dominant equations and

hence obtain an approximation to the exact solution. Start with trial

solution

w= 0.0, x= 0.0, y= 0.0, z= 0.0 (15 Marks)

Multiply out first line to find wSolving row 1 for w;

Rearranged this gives;

Solving row 2 for x;



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The resultant values can now be used in the next cycle, note: because Im usingthe

Gauss-Seidel iteration scheme the values are updated as they are calculated foreach equation;


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Summary of results

w x y z

Cycle using given figures 0 0 0 0

1st cycle -0.05 -0.987 1.02 1.002

Results of 1st cycle -0.0014 -1 0.999 1

Resultant Values 0 -1 1 1

I shall now replace w, x, y, z with the resultant values

It can be seen that convergence has been achieved and that the resultant valueshave been proved to be accurate.


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(c) Explain the difference between Gauss-Seidel and Jacobi methods.

(5 Marks)

The difference between the Gauss-Seidel and Jacobi methods is that the Gauss-

Seidal method involves updating the equations as soon as a new component (x, y, z

etc) value has been calculated during the cycle. The Jacobi method only uses the full

cycle values for each of the component values i.e. once a full cycle has been

completed and values are found for all components. This makes the Gauss-Seidel

method far quicker to use in order to get the required values.

Question 2

(a) Perform a Triangular Decomposition on the symmetrical Matrix; (10 Marks)

Triangular decomposition involves splitting this symmetrical Matrix (A) into twocomponent parts a Lower (L);

As well as an Upper (U);

It follows that the symmetrical matrix (A) equals the product of the componentparts (L and U) giving us;

A = LxU

= Multiplication of the matrices (L U) gives the following results;


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Having multiplied the matrices I can now calculate the component values bysolving the resultant equations;


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The component values can now be inserted into the equation , this will complete thetriangular decomposition;

A = L x U

= =

(b) Use your result from (a) above to solve the symmetrical system of linearequations. (10 Marks)

=

A X B

The above symmetrical system of linear equations can be expressed as;

I have proved above that;

=

The equation can now be expressed as;

=

In order to complete the calculation I will create new matrix Y, it is equal to theproduct of matrices U and X;


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=

Therefore;

I will now calculate the components of Y as follows;

The components of the Y matrix can now be used to calculate the components of X;

And since then;


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Multiplication of the matrices (U X) gives the following results;

I shall now replace x, y, z with the resultant values

It can be seen that convergence has been achieved and that the resultant valueshave been proved to be accurate.

(c) Briefly discuss the methods you would use for solving various types of

systems of linear equations. (5 Marks)

Systems of linear equations are linear equations that need to be solved

simultaneously, the methods I would use would depend on the type of system. If thesystem of linear equations was small and dense (i.e. packed with numbers) I would

use a direct method such as the Inverse method or Triangular Decomposition.

Matrix Inverse Method

This method involves using the inverse of Matrix A (A-1). The inverse matrix is

calculated by finding the determinant and the adjoint. The adjoint is divided by the

determinant and the resultant inverse matrix (A-1) is then multiplied to both sides of

the system;

A-1 x = A-1x B


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X = A-1 x B

Values for X are then used for the initial condition;

A x X = B

Triangular Decomposition

This method involves creating two new matrices an upper (U) and a lower matrix (L)

from the initial Matrix (A), U and L are used to find a new matrix (Y) which can in turn

be used to find matrix X components;

A x X = B (original condition)

U x L x X =B ( lower and upper forms)

L x Y = B ( Y can be calculated)

U x X = Y (X can now be calculated using the values calculated for Y)

If The matrix is larger and is more sparse i.e. has many zeros or small values then

an indirect method such as Gauss-Seidel of Jacobi iteration scheme would be

used.

Gauss-Seidel and Jacobi Methods

Both of these method start by changing the matrix( A) so that it is diagonally

dominant ( this guarantees convergence). The figures for components in the X matrix

are then estimated and from the resultant equations the true figures are calculated.

The Gauss-Seidel method involves updating the equations as soon as a new

component (x, y, z etc) value has been calculated during the cycle. The Jacobi

method only uses the full cycle values for each of the component values i.e. once a

full cycle has been completed and values are found for all components. This makes

the Gauss-Seidel method far quicker to use in order to get the required values.

Question 3

(a) Evaluate the determinant of the matrix: (3 Marks)

In order to evaluate the determinant I must first choose a pivot point then calculate a

scalar matrix, the pivot point is shown in blue the matrix components are shown inred;


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A =

Giving;

The rest of the matrix is calculated in a similar way;

det(A) = a (e i - h f) - d (b i - h c) + g (b f - e c)

det(A)= The determinant of matrix A can now be calculated

det(A)= 2 - 0 + 0det(A) = 2 (-1-1)det(A) = - 4(b) Using your result from (a) above calculate the inverse of (10 Marks)

Determinant from previous calculation;

det(A) = -4

Now I will Form Cofactor of Matrix A (coA);


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It can be seen that equality has been achieved and that the inverse values have

been proved to be accurate.

(c)From the results from (a) and (b) above Solve (6Marks)

=

In order to solve this set of linear equations the inverse matrix (A-1) is then multiplied

to both sides of the system;

A-1 x = A-1x BThis can then be simplified to;

X = A-1 x B

Values for X can now be used for the initial set of equations;


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It can be seen that equality has been achieved and that the x, y z values of matrix Xhave been proved to be accurate.

(d)Evaluate the double integral (6 Marks)

In order to evaluate the double integral I will first integrate with respect to y,

Now I need to integrate with respect to x;


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Question 4

(a) A dynamical engineering system has 2 time (t) dependent generalised

coordinates X,Y which obey the simultaneous differential equations;

=

Verify that a specific solution to the above equation is

= a exp (1t) 1 + b exp (2t) 2Where a, b are time independent coefficients. 1, 2 are Eigen values and 1, 2are normalised eigenvectors of the matrix;

(12 Marks)

Inputting the Eigen values and eigenvectors will give me;

Now I must differentiate the expressions; a + b a + b

On the other side of the equation Ill have:

= [a + b ]


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= a + b = a + b

This confirms of the specific solution to the above equation is:

= a + b (b) The matrix in (a) above

has eigenvalues 0, 2 and normalised eigenvectors 1, 2 given by;

Use this result for the eigenvalues and normalised eigenvectors to calculate

the coefficients a, b for the initial condition;

= (13 Marks)I need to find the secular determinant (Det) as follows;

I shall now check the values using the trace (Tr) method;


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Now I can find the 2nd Eigen Vector

This gives me the Ratio of the Eigen vector of 1:1

In order to normalise the Eigen vector I must scale by a factor of S


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Where:

Table of

1 0 0 1

Coefficient a(0)


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Coefficient b(0)

Initial condition coefficients

a(0) and b(0)

Question 5

(a) Obtain the general solution to the differential equation. (10 Marks)+ + y =0In order to obtain a general solution for this differential equation I will use the trialsolution method;


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I have supposed that

Now I must substitute;

The general solution for the differential equation is therefore;

(b) Use the techniques of Laplace transformation to solve the differential

equation

+ + 2y =0, = 3, y(0) =4 (15 Marks)


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Therefore;

Question 6

(a) By deriving an expression for sverify that the Fourier Series Is a solution to the partial differential equation

+

(10 marks)

Left Hand Side


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Right Hand Side

This therefore verifies that the Fourier equation is a solution to the PDE.(b) Set up the finite difference equations to numerically estimate a solution

for the temperature at (a,b,c) to the Laplace equation+ = 0 for a sheet of metal with the boundary conditions shownbelow (8 marks)


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The function at steady state;

Temperature at point A;

Temperature at point B;

Temperature at point C; Diagonally Dominant Matrix;

From the diagonally dominant matrix I can now calculate the following;

Temperature at point A;

Temperature at point B;

Temperature at point C;


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Therefore;

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(c) Solve your equations iteratively performing 2 cycles of your iteration

scheme

Taking the guesses for the temperatures at (a,b,c) all as 25 degrees

centigrade (7 marks)

I shall use the Gauss-Seidal iteration scheme to solve the set of equations;

First cycle

Second cycle


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Surface area of the box has the following constraint;

Lagrange equations

b) Solve your equations for the optimal values of s and h

Therefore;

I can now combine and simplify the equations;

Therefore;

This would indicate that the height would need to be the same length as the sides;therefore the box would have to be a cube and as such will have square surfacesinstead of rectangular.

Length of box side;


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33At the point (x,y,z)=(1,2,3) the function A is not solenoidal. I can determinethis due to that fact that there is divergence occurring (-33), in order for A tobe Solenoidal at this point it wound need to be divergenceless1 or in otherwords the divergence wound need to be equal to zero.

(c) If B = xi + y j , sketch the vector field B (6 marks)

Point (X,Y) V (X,Y)Direction of vector From

(X,Y) Co-ordinate

(1,1) =1i + 1j 1,1

(1,-1) =1i - 1j 1,-1

(-1,-1) =-1i -1j -1,-1

(-1,1) =-1i + -1j -1,1

(2,1) =2i + 1j 2,1

(1,2) =1i + 2j 1,2

(2,-1) =2i - 1j 2,-1

(1,-2) =1i - 2j 1,-2

(-2,-1) =-2i - 1j -2,-1

(-1,-2) =-1i -2j -1,-2

(-2,1) =-2i - 1j -2,1

(-1,2) =-1i - 2j -1,2

1Advanced electromagnetism: foundations, theory and applications

By Terence William Barrett, Dale M. Grimes


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I can now plot the points;

Y

X

(d) Find curl B for the vector field B defined in (c) above. Is B

irrotational? (7 marks)

Curl is a function of (x,y,z)

B=xi + yj

The fact that the curl is equal to zero would indicate that the vector field isirrotational (conservative)


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Past paper 2

Question1

(a) The fluid levels h1 and h2 in two tanks are computer controlled to

obey the pair of simultaneous differential equations;

= Where t is the time. By calculating the eigenvalues and eigen vectors

of an appropriate matrix verify that a general solution to the above

equation is

Where p,q are time independent co-efficients. (18 marks)

I need to find the secular determinant (Det) as follows;


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I shall now check the values using the trace (Tr) method;

= -2 The Trace confirms that the calculations are correct

Now I can find the 1st Eigen Vector

Input the values calculated from the secular determinant

Multiplication of the matrices gives

The gives me a ratio of the Eigen vectors of 1:1 In order to normalise the Eigen vector I must scale by a factor of S


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Now I can find the 2nd Eigen Vector

This gives me the Ratio of the Eigen vector of 1:1

In order to normalise the Eigen vector I must scale by a factor of S


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(a) Use your result for the eigenvalues and normalised eigenvectors to

calculate the coefficients p and q for the initial condition (t = 0)

(7 marks)

Table of

1 0 0 1

Coefficient p(0)


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Coefficient q(0)

Initial condition coefficients are the same

Question 2

(a) Perform a Triangular Decomposition on the symmetrical Matrix; (10 Marks)

Triangular decomposition involves splitting this symmetrical Matrix (A) into twocomponent parts a Lower (L);


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As well as an Upper (U);

It follows that the symmetrical matrix (A) equals the product of the component parts(L and U) giving us;

A = LxU

= Multiplication of the matrices (L U) gives the following results;


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Having multiplied the matrices I can now calculate the component values by solvingthe resultant equations;

The component values can now be inserted into the equation , this will complete thetriangular decomposition;

A = L x U

= =


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The components of the Y matrix can now be used to calculate the components of X;

And since then;

Multiplication of the matrices (U X) gives the following results;

I shall now replace x, y, z with the resultant values

It can be seen that convergence has been achieved and that the resultant values

have been proved to be accurate.


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(c) Briefly discuss the methods you would use for solving various types of

systems of linear equations. (5 Marks)

Systems of linear equations are linear equations that need to be solved

simultaneously, the methods I would use would depend on the type of system. If the

system of linear equations was small and dense (i.e. packed with numbers) I would

use a direct method such as the Inverse method or Triangular Decomposition.

Matrix Inverse Method

This method involves using the inverse of Matrix A (A-1). The inverse matrix is

calculated by finding the determinant and the adjoint. The adjoint is divided by the

determinant and the resultant inverse matrix (A -1) is then multiplied to both sides of

the system;

A-1

x = A-1

x B

X = A-1 x B

Values for X are then used for the initial condition;

A x X = B

Triangular Decomposition

This method involves creating two new matrices an upper (U) and a lower matrix (L)

from the initial Matrix (A), U and L are used to find a new matrix (Y) which can in turn

be used to find matrix X components;

A x X = B (original condition)

U x L x X =B ( lower and upper forms)

L x Y = B ( Y can be calculated)

U x X = Y (X can now be calculated using the values calculated for Y)

If The matrix is larger and is more sparse i.e. has many zeros or are small then an

indirect method such as Gauss-Seidel of Jacobi iteration scheme would be used.

Gauss-Seidel and Jacobi Methods

Both of these method start by changing the matrix( A) so that it is diagonally

dominant ( this guarantees convergence). The figures for components in the X matrix

are then estimated and from the resultant equations the true figures are calculated.

The Gauss-Seidel method involves updating the equations as soon as a new

component (x, y, z etc) value has been calculated during the cycle. The Jacobi


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method only uses the full cycle values for each of the component values i.e. once a

full cycle has been completed and values are found for all components. This makes

the Gauss-Seidel method far quicker to use in order to get the required values.

Question 3

(a) Arrange the equations below into diagonally dominant form giving a

motivation for doing so.

(8 Marks)

=

By arranging the matrix into a diagonally dominant form convergence is guaranteed,

when using both the Jacobi and the Gauss-Seidel iteration schemes, without

arranging the matrix into diagonally dominant form the equation may not converge.

=

(b) Derive an expression for the Gauss-Seidel or Jacobi iteration scheme in

matrix form.

(8 Marks)

An expression for the Gauss-Seidel or Jacobi iteration scheme can be derived fromthe above set of equations;

Multiply out first line to find wSolving row 1 for w;



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Solving row 2 for x;


Solving row 3 for y;


Solving row 4 for z; Rearranged this gives;

(c) Perform one cycle of the Gauss-Seidel iteration scheme for solving a

system of linear equations on your diagonally dominant equations and

hence obtain an approximation to the exact solution. Start with trial

solution w= 0.0, x= 0.0, y= 0.0, z= 0.0 (12 Marks)

I will now start by using the given trial solution in the above rearranged equations;If w = y = x = z = 0

Then;


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The resultant values can now be used in the next cycle, note: because Im using theGauss-Seidel iteration scheme the values are updated as they are calculated foreach equation;


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Summary of results

w x y z

Cycle using given figures 0 0 0 0

1st cycle -0.05 0.99 -1 0.1

Results of 1st cycle -0.0045 0.999 -1.001 -0.005

Resultant Values 0 1 -1 0


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I shall now replace w, x, y, z with the resultant values

It can be seen that convergence has been achieved and that the resultant values


Question 4

(a) Multiply the matrices

(5 marks)

(b) If Then use the turn over rule to evaluate

(5 marks)

The following formula applies to the turn over rule


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(c) Calculate the determinant of;

(5 marks)In order to evaluate the determinant I must first choose a pivot point then calculate ascalar matrix, the pivot point is shown in blue the matrix components are shown inred;

M = Giving;

The rest of the matrix is calculated in a similar way;

Determinantof matrix m

(d) Use your result for (c) to evaluate the inverse of;

(5 marks)


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Now I will Form Cofactor of Matrix M (coM);

I must now transpose the cofactor matrix to produce the adjoint, in this case theadjoint is same (as the cofactor matrix is symmetrical it will not be affected bytransposition)

In order to find the inverse of the matrix the following formula is used;

To verify the correctness of the inverse matrix


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It can be seen that equality has been achieved and that the inverse values havebeen proved to be accurate.

Therefore;

The inverse of

(e) Use your result from (c) to solve the simultaneous linear equations

(5 marks)In order to solve this set of linear equations the inverse matrix (M-1) is then multiplied

to both sides of the system;

M-1 x = M-1x BThis can then be simplified to;

X = M-1 x B

Values for X can now be used for the initial set of equations;


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It can be seen that equality has been achieved and that the x, y z values of matrix X


Therfore;

Section B

Question 5

(a) The vibration of a mass (m) on the end spring with stiffness constant k

and damping constant obeys the differential equation; + + kx =0Find the general solution to the differential equation above. (10 marks)

In order to obtain a general solution for this differential equation I will use the trialsolution method;

I have supposed that


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The general solution for the differential equation is therefore;

(b) Use the technique of laplace transformation to solve the differential

equation

given y(0) = 0 (15 marks)A table of Laplace transforms are given in the appendix

LAPLACE TRANSFORMATION

From Laplace transforms table;

Given;


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Therefore;

Question 6

A manufacturer wants to make a tin can (radius r and height h) with the

largest possible volume out of exactly 1 metre squared of material of

negligible thickness.

(a) Using Lagrange undetermined multipliers derive 3 equations describing

the above optimisation problem. (15 marks)

R

h

Volume of the box can be found using the following equation; Surface area of the box has the following constraint;

Lagrange equations


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(b) Solve your equations for the optimal values of r and h. (10 marks)

Therefore;

Therefore;

I can now combine and simplify the equations;

This would indicate that the height would need to be twice the length of the radius.

To calculate the radius;

Because I have calculated that;


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To calculate the Height;

Optimised Radius and height;

Area check;

Rectangular component = Height x circumference = 0.46 x (0.46 x ) = 0.664 mCircular area = x = 0.333mTotal area = 0.664 x 0.333 = 0.997 so 99.7% of the allowed material has been used.


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Question 8

(a) Fluid in an oil reservoir is diffusing through a two-dimensional slab of

porous rock. In the steady state the concentration C(x,y) obeys the

Laplace Equation;

+ = 0 (10 marks)Set up the finite difference equations to numerically estimate a solution

for the concentration of fluid at the points

for the rock slab with

the boundary conditions shown below; (10 marks)

Y

Equations for calculating Temperature in the X Direction;


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Equations for calculating Temperature in the Y Direction;

Equations for calculating Temperature in the X and Y Direction;

If x=y then;

The function at steady state; Temperature at point ; Temperature at point

;


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Temperature at point ;

Therefore;

Diagonally Dominant Matrix;

From the diagonally dominant matrix I can now calculate the following;




(b) Sollve your equations iteratively performing 1 cycle of your iteration

scheme taking the initial values for the concentrations at as 20,30, 40 concentration units respectively. (5 marks)

shall use the Gauss-Seidal iteration scheme to solve the set of equations;


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First cycle

(c) Verify that the Fourier series

Is a solution of the wave equation given by

(10 marks)(x is the position and s and c are constants, j = )

Ta Tb Tc

Cycle using given figures 20 30 40

1st cycle 23.13 33.28 40.82

Resultant Values 23 33 41


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Bibliography

Advanced electromagnetism: foundations, theory and applications

By Terence William Barrett, Dale M. Grimes

Essentials of engineering mathematics

By Alan Jeffery

Engineering Mathematics 6th Edition

By K.A Stroud

Documents

Dougs Engineering Mathmatics Cousework for Semester 1 Final