MM140 Engineering Mathematical Methods Assignment Sample NUIG Ireland

The MM140 Engineering Mathematical Methods course covers a range of mathematical techniques that are commonly used in engineering. These techniques include matrix algebra, eigenvalues and eigenvectors, numerical analysis, ordinary and partial differential equations, and scientific computing.

The course is designed to provide students with the necessary skills and knowledge to solve problems in engineering. It also provides students with a strong foundation in mathematics, which can be used for further studies or employment in the engineering field.

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In this section, we are describing some assigned activities. These are:

Assignment Activity 1: Express a problem modelled by a system of linear equations in an appropriate matrix form and solve the resulting system of equations.

The problem modelled by a system of linear equations can be expressed in matrix form as follows:

Ax = b, where A is an n x n matrix, x is an n-element column vector, and b is an n-element row vector. The system of linear equations can be solved by using the Gauss-Jordan elimination method, which results in the following reduced Row Echelon Form (RREF) for the matrix A:

A = [I 0 ··· 0] · [D −1 0 ··· 0] [A′] · [x],

where I is an n x n identity matrix, D is a diagonal matrix with the diagonal elements equal to the reciprocals of the eigenvalues of A, and A′ is an n x n matrix with the eigenvectors of A as its columns. The solution to the system of linear equations is then given by x = A′−1b.

Assignment Activity 2: Use row operations to determine whether or not a system of m linear equations in n unknowns is consistent/has a unique solution /has an infinite number of solutions.

There are three different possible outcomes when solving a system of linear equations: the system is consistent and has a unique solution, the system is consistent but has an infinite number of solutions, or the system is inconsistent. To determine which outcome applies to a given system, we can use row operations.

Row operations involve adding or subtracting one row from another, or multiplying a row by a constant. These operations do not change the solution set of the linear system; they simply provide a way to simplify the equations while preserving their solution set. In particular, we can use row operations to transform an inconsistent system into a consistent one with no solutions, or vice versa. 

To see how this works, consider the following example:

$$\begin{aligned}

x+y&=1\\

x-y&=2

\end{aligned}$$

This system is inconsistent; adding the two equations together gives $2x=3$, which has no solutions. However, we can use row operations to transform this system into a consistent one:

$$\begin{aligned}

x+y&=1\\

-(x-y)&=-2

\end{aligned}$$

Now, subtracting the second equation from the first gives $2y=3$, which has a unique solution of $y=\frac{3}{2}$ and $x=\frac{5}{2}$.

In general, a system of linear equations is consistent if and only if the reduced Row Echelon Form (RREF) of the coefficient matrix has at least one non-zero row. A system of linear equations is inconsistent if and only if the RREF of the coefficient matrix has all zero rows.

Assignment Activity 3: Perform elementary calculations involving matrices and determinants.

Matrix multiplication is a way of performing calculus on matrices that allows one to determine and use students’ performance on exams, find new theorems, and solve systems of linear equations. If you have two matrices A and B, you can multiply them together to get a third matrix C = AB. The process is quite simple. All you need to do is take the dot product of each row in matrix A with each column in matrix B. So, if A is a 2×3 matrix and B is a 3×4 matrix, then C will be a 2×4 matrix.

To calculate the dot product, you simply multiply the corresponding elements in each row/column and add them up. So, for the first element in C, you would take the dot product of the first row in A with the first column in B. For the second element, you would take the dot product of the first row in A with the second column in B, and so on.

Assignment Activity 4: Calculate the characteristic polynomial, eigenvalues, and corresponding eigenvectors for a 3 x 3 matrix, and diagonalize such a matrix.

The characteristic polynomial, sometimes called the moderated polynomial, of a 3×3 matrix, is a certain 9th degree polynomial in the indeterminants x, y, and z of the form:

c(x,y,z) = det(A – xI).

Eigenvalues are the roots (in any field extension of the ground field) of this equation. For example, if F is the field of real numbers, then an eigenvalue could be 2.14159…γ ∈ C (the set of all complex numbers). The corresponding eigenvector would be a non-zero vector v such that Av = λv. In other words, if you took this 3×3 matrix, multiplied it by some vector, and the result was that the vector got stretched by a factor of λ (the eigenvalue), then that vector would be an eigenvector.

Eigenvectors are not always easy to find; for most people, they are quite difficult. However, there is a method that can be used to calculate them, known as the characteristic equation. This equation is derived from the fact that an eigenvector must satisfy the equation:

A – λI = 0,

where I am the 3×3 identity matrix and λ is an eigenvalue of A.

To find the characteristic equation, you simply take the determinant of this matrix equation:

det(A – λI) = 0.

This will give you a 9th-degree polynomial in λ, which is known as the characteristic equation. The roots of this equation are the eigenvalues of A. To find the corresponding eigenvectors, you simply plug the eigenvalues back into the original matrix equation and solve for the unknown vector.

Assignment Activity 5: Write complex numbers in modulus/argument form, apply de Moivre’s theorem, derive expressions for the sin/cosine of multiple angles in terms of powers of sin/cosine x, etc.

Complex numbers can be written in modulus/argument form, where the modulus is the length of the vector and the argument is its angle. This can be easily seen by rotating a vector on the complex plane by 90 degrees:

Multiplying a complex number by I rotates it by 180 degrees, or in radians, 2pi. So if we want to find the cosine of 45 degrees, for example, we just need to multiply it by i:

Similarly, we can find the sin of an angle using de Moivre’s theorem: where m is an integer. So for example, to find sin(3*pi/4), we just need to use De Moivre’s theorem with m=1:

We can also derive expressions for the sin and cosine of multiple angles in terms of the powers of sin and cosine. 

Assignment Activity 6: Factorize real polynomials into irreducible linear and quadratic terms. Determine the nth roots of unity for small values of n.

A real polynomial can be factorized into irreducible linear and quadratic terms if and only if the degree of the polynomial is even. The nth roots of unity can be used to determine whether or not a polynomial is irreducible. If the nth roots of unity are all distinct, then the polynomial is irreducible. If one or more of the nth roots of unity are equal, then the polynomial is reducible.

For example, consider the following polynomial:

x^4 – 2x^2 – 1

The nth roots of unity for this polynomial are 1, i, −I, and −1. As you can see, two of the roots are equal, so the polynomial is reducible.

Assignment Activity 7: Plot direction fields for first order Ordinary Differential Equations (ODEs) and solve separable first-order ODEs.

There are a few different ways to plot direction fields for first-order ODEs. One way is to use a graphing calculator or computer software like Wolfram Alpha. Another way is to manually plot the points and connect them with tangent lines. This can be tedious, but it can give you a better feel for what the direction field is telling you about the behaviour of the solutions to the ODE.

To get started, let’s review how to solve separable first-order ODEs. These are differential equations of the form: dy/dx = f(y)/g(x). To solve these equations, we need to find an equation for y as a function of x that satisfies this condition. We can do this by first separating the variables:

dy/dx = f(y)/g(x)

dy = f(y) dx/g(x)

Then we need to integrate both sides:

∫ dy = ∫ (f(y) dx)/g(x)

y = ∫ (f(y) dx)/g(x) + C

Finally, we need to solve for y:

y = ∫ (f(y) dx)/g(x) + C

Assignment Activity 8: Solve linear first-order ODEs by the integrating factor method.

To solve a linear first-order ODE by the integrating factor method, first, calculate the integrating factor. This is done by finding the derivative of each term in the equation and multiplying each term by the integrating factor..”;

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Assignment Activity 9: Solve linear homogeneous second-order ODEs with constant coefficients, solve linear non-homogeneous second-order ODEs with constant coefficients by the method of undetermined coefficients and the method of variation of parameters.

A linear homogeneous second order ODE with constant coefficients has the form:

a_2 y” + a_1 y’ + a_0 y = 0

To solve this type of differential equation, we need to find a general solution of the form:

y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x}

where r_1 and r_2 are the roots of the characteristic equation:

a_2 r^2 + a_1 r + a_0 = 0

Once we have the general solution, we can then find the particular solution by plugging in the initial conditions.

A linear non-homogeneous second order ODE with constant coefficients has the form:

a_2 y” + a_1 y’ + a_0 y = f(x)

To solve this type of differential equation, we need to find a general solution of the form:

y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x} + y_p(x)

where r_1 and r_2 are the roots of the characteristic equation:

a_2 r^2 + a_1 r + a_0 = 0

and y_p(x) is a particular solution of the form:

y_p(x) = y_0 + y_1 x + y_2 x^2 + …

Once we have the general solution, we can then find the particular solution by plugging in the initial conditions.

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