Mathematics Revision Materials

Mathematics Revision Materials#

Introduction#

Your course lectures will assume you arrive with a certain amount of background in mathematics. Key topics that the programme will build on are Linear Algebra, Calculus, Differential Equations, Vector Calculus and Statistics. If you are unfamiliar with any of these topics, or if it has been a while since you last interacted with them, you shoudl try to brush up your skills before October. You can check and test your understanding with the sample questions and answer provided in the tabs below .

Sample questions (click on the answers to reveal them):

  1. If \(A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\) and \(B = \begin{bmatrix} 3 & 2 \\ 1 & 2 \end{bmatrix}\), what is \(AB - BA\)?

    Answer
    \[\begin{split} AB-BA = \begin{bmatrix} -2 & -2 \\ 2 & 2 \end{bmatrix}\end{split}\]

    If you struggled with this question, you may want to revise matrix multiplication.

  2. Find the solution (or solutions) to the following matrix equation:

    \[\begin{split} \begin{bmatrix} 1 & 3 \\ 2 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 10 \end{bmatrix} \end{split}\]
    Answer

    The matrix system is singular, so there is no unique solution. The second row is a multiple of the first row, so the system is underdetermined. The solution space is a line in \(\mathbb{R}^2\), given by the equation \(y = -\frac{1}{3}x+ \frac{5}{3} \).

    If you struggled with this question, you may want to revise matrix determinants.

  3. What is the inverse of the matrix \(A = \begin{bmatrix} 1 & 2 & 1\\ 2 & 2 &1 \\ 0 & 0& 2\end{bmatrix}\)?

    Answer
    \[\begin{split}A^{-1}=\begin{bmatrix} 0.5 & -0.5 & 0 \\ -1 & 1 & 1 \\ 0 & 0 & 0.5 \end{bmatrix}\end{split}\]

    If you struggled with this question, you may want to revise matrix determinants and matrix inverses.

  1. Find the eigenvalues and eigenvectors of the following matrix:

    \[\begin{split} A = \begin{bmatrix} 1 & 3 & 0\\ 3 & 1 & 0\\ 0 & 0 & 3 \end{bmatrix}\end{split}\]
    Answer

    Eigenvalues: \(\lambda_1 = 3, \lambda_2 = 2, \lambda_3 = 4\)

    Eigenvectors: \(v_1 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, v_2 = \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}, v_3 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}\)

    (or any scalar multiple of these vectors).

    If you struggled with this question, you may want to revise eigenvalues and eigenvectors.

Revision material:

Sample questions (click on the answers to reveal them):

  1. Find the derivative of the function \(f(x) = x^2 + 3x + 2\tan(2x)+7\).

    Answer
    \[ f'(x) = 2x + 3 + 4\sec^2(2x)\]

    If you struggled with this question, you want want to revise the rules of differentiation.

  2. Find the integral of the function \(f(x) = 2x + 3\).

    Answer
    \[ \int f(x) dx = x^2 + 3x + c\]

    where \(c\in \mathbb{R}\) is the constant of integration.

  3. Find the maximum and minimum values of the function \(f(x) = x^3 - 3x^2 + 2\) on the interval \([-1, 3]\).

    Answer

    Maximum at \(x = 0\), minimum at \(x = 2\)

Revision material:

Sample questions (click on the answers to reveal them):

  1. Find a general solution for the following ordinary differential equation:

    \[\frac{d^2 y}{dx^2} + 3 \frac{dy}{dx} + 2y = \cos(x)\]
    Answer
    \[ y = c_1 e^{-x} + c_2 e^{-2x} + \frac{1}{5} \cos(x) + \frac{3}{5} \sin(x)\]

2 Find the trajectory of the following first-order differential equation:

\[\frac{dy}{dx} = y^2 - 1\]
\[ y(1)= \frac{1}{2} \]
Answer
\[ y = \frac{1}{3 - x}\]

Revision material:

Sample questions (click on the answers to reveal them):

  1. Find the gradient of the function \(f(x, y) = x^2 + y^3\) at the point \((1,2)\).

    Answer
    \[\begin{split} \left. \nabla f \right|_{(1,2)} = \begin{bmatrix} 2 \\ 12 \end{bmatrix}\end{split}\]

  2. Find the partial derivative of the function \(f(x, y, z) = x^2 + y^3 + z^4 + \sin(xz)\) with respect to \(z\).

Answer
\[ \frac{\partial f}{\partial z} = 4z^3 + x\cos(xz) \]

  1. Find the rate of change of the vector function \(\mathbf{r}(t) = \begin{bmatrix} t^2 \\ 2t \\ \cosh(3t) \end{bmatrix}\) with respect to \(t\).

    Answer
    \[\begin{split} \frac{d\mathbf{r}}{dt} = \begin{bmatrix} 2t \\ 2 \\ 3\sinh(t) \end{bmatrix}\end{split}\]

  2. Find the total derivative of the function \(f\) with respect to \(t\) where

    \[ f(x(t), y(t)) = x^2 + \tan(y) \]
    \[ x(t) = 2t, y(t) = 3t \]
    Answer
    \[ \frac{df}{dt} = 4t + 3\sec^2(3t)\]

Revision material:

Sample questions (click on the answers to reveal them):

  1. On a 10-sided die, what is the probability of rolling a number divisible by 3?

    Answer
    \[ P(x \text{ is divisible by 3}) = \frac{3}{10}\]

  2. If the probability of rain on any given day is 0.3, what is the probability of no rain for 3 consecutive days?

    Answer
    \[ P(\text{no rain for 3 days}) = 0.7^3 \approx 0.34\]

  3. If the chance Maria will make pancakes for breakfast is 1/7, the chance she will have noodles for dinner is 1/3 and these chances are independent, what is the probability she will have pancakes for breakfast and noodles for dinner?

    Answer
    \[ P(\text{pancakes and noodles}) = \frac{1}{7} \times \frac{1}{3} = \frac{1}{21}\]

Revision material:

Sample questions (click on the answers to reveal them):

  1. Trigonometry: rewrite the following expression in terms of \(\sin(x)\) and \(\cos(x)\):

    • \(\tan(x)\)

    • \(\sin(3x)+cos(2x)\)

    • \(\sec^2 (x)\)

    Answer
    \[ \tan(x) = \frac{\sin(x)}{\cos(x)}\]
    \[ \sin(3x) + \cos(2x) = 3\sin(x)\cos^2(x) - 2\sin^2(x) + \cos^2(x) - \sin^2(x)\]
    \[ \sec^2(x) = \frac{1}{\cos^2(x)}\]

  2. Geometry: find the area of a triangle with sides of length 5, 5 and 8.

    Answer

    The triangle is isoceles, with two sides of the same length. This means its area is twice that of a right -angled triangle with hypotenuse 5 and one size 8/2=4. The height is thus \(\sqrt{5^2 - 4^2} = 3\). The area is thus \(\frac{1}{2} \times 8 \times 3 = 12\).

  3. Algebra: what are the roots of the quadratic equation \(x^2 - 5x + 6 = 0\)?

    Answer

    The roots are \(x = 2\) and \(x = 3\).

Additional Learning Resources#

You may find the following addtional resources useful to get up to speed before the course starts:

  1. Imperial Metric is an online interactive Maths resource aimed at first-year undergraduates covering high school material, but it may be useful for brushing up on topics learned some time ago. The resource can be accessed here.

    We particularly recommend reviewing any of the following topics that you do not feel confident with:

    • Linear Algebra

      • Matrices

      • Vectors

    • Statistics and probability

    • Calculus

      • Differentiation

      • Integration

      • Partial differentiation & vector calculus

    • Differential equations

    • Numerical methods

  2. A concise set of lecture notes introducing Vector Calculus and PDEs from the undergraduate maths courses in the Department of Earth Science and Engineering are available for download here

  3. More general mathematics materials for all years of undergraduate tuition from the Earth Science and Engineering department at Imperial College are available here. Many of these notes also provide coding examples using the techniques in Python.

External Learning Resources#

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