A-level Pure Maths 2&3

(In preparation...)
We continue our review of the A-level pure maths as an introduction to university-level mathematics. We are NOT interested in the A-level exams. We are interested in refining and deepening our understanding of the maths!
  •  Lecture 1. Algebra
    1. Operations with polynomials
    2. Solutions of polynomial equations
    3. The modulus function
  • Lecture 2. Logarithms and exponentials
    1. Logarithms
    2. Exponential functions
    3. Modeling curves
    4. The natural logarithm function
    5. The exponential function
  • Lecture 3. Trigonometry
    1. Reciprocal trigonometric functions
    2. Compound-angle formulae
    3. Double-angle formulae
    4. The forms \(r \cos(\theta \pm \alpha)\), \(r\sin(\theta \pm \alpha)\)
    5. The general solutions of trigonometric equations
  • Lecture 4. Differentiation (Part 1)
    1. The product rule
    2. The quotient rule
    3. Differentiating natural logarithms and exponentials
    4. Differentiating trigonometric functions
  • Lecture 5. Differentiation (Part 2)
    1. Differentiating functions defined implicitly
    2. Parametric equations
    3. Parametric differentiation
  • Lecture 6. Integration
    1. Integrals involving the exponential function
    2. Integrals involving the natural logarithm function
    3. Integrals involving trigonometric functions
    4. Numerical integration
  • Lecture 7. Further algebra
    1. The general binomial expansion
    2. Algebraic functions
    3. Partial fractions
    4. Using partial fractions with the binomial expansion
  • Lecture 8. Further integration (Part 1)
    1. Integration by substitution
    2. Integrals involving exponentials and natural logarithms
    3. Integrals involving trigonometrical functions
  • Lecture 9. Further integration (Part 2)
    1. The use of partial fractions in integration
    2. Integration by parts
    3. General integration
  • Lecture 10. Differential equations
    1. Forming differential equations from rates of change
    2. Solving differential equations
  • Lecture 11. Vectors (Part 1)
    1. The vector equation of a line
    2. The intersection of two lines
    3. The angle between two lines
    4. The perpendicular distance from a point to a line
  • Lecture 12. Vectors (Part 2)
    1. The vector equation of a plane
    2. The intersection of a line and a plane
    3. The distance of a point from a plane
    4. The angle between a line and a plane
    5. The intersection of two planes
  • Lecture 13. Complex numbers (Part 1)
    1. The growth of the number system
    2. Working with complex numbers
    3. Representing complex numbers geometrically
    4. Sets of points in the complex plane (the Argand diagram)
  • Lecture 14. Complex numbers (Part 2)
    1. The polar form of complex numbers
    2. Sets of points using the polar form
    3. Working with complex numbers in polar form
    4. Complex exponents
    5. Complex numbers and equations

Reference

  • Sophie Goldie (2012), Pure Mathematics 2 and 3, Hodder Education.
  • Jean Linsky, James Nicholson, Brian Western (2018), Complete Pure Mathematics 2/3, Oxford University Press.

Comments

Post a Comment

Popular posts from this blog

Open sets and closed sets in \(\mathbb{R}^n\)

Image
Open sets In \(\mathbb{R}\), we have the notion of an open interval such as \((a, b) = \{x \in \mathbb{R} | a < x < b\}\). We want to extend this idea to apply to \(\mathbb{R}^n\). We also introduce the notions of bounded sets and closed sets in \(\mathbb{R}^n\). Recall that the \(\varepsilon\)-neighbor of a point \(x\in\mathbb{R}^n\) is defined as \(N_{\varepsilon}(x) = \{y \in \mathbb{R}^n | d(x, y) < \varepsilon \}\) where \(d(x,y)\) is the distance between \(x\) and \(y\). Definition (Open set) A subset \(U\) of \(\mathbb{R}^n\) is said to be an  open set  if the following holds: \[\forall x \in U ~ \exists \delta > 0 ~ (N_{\delta}(x) \subset U).\tag{Eq:OpenSet}\] That is, for every point in an open set \(U\), we can always find an open ball centered at that point, that is included in \(U\). See the following figure. Perhaps, it is instructive to see what is  not  an open set. Negating (Eq:OpenSet), we have \[\exists x \in U ~ \forall \delta > 0 ~ (N_{\delta}(x) \not
Read more

Euclidean spaces

Image
We would like to study multivariate functions (i.e., functions of many variables), continuous multivariate functions in particular. To define continuity, we need a measure of "closeness" between points. One measure of closeness is the Euclidean distance. The set \(\mathbb{R}^n\) (with \(n \in \mathbb{N}\)) with the Euclidean distance function is called a Euclidean space. This is the space where our functions of interest live. The real line is a geometric representation of \(\mathbb{R}\), the set of all real numbers. That is, each \(a \in \mathbb{R}\) is represented as the point \(a\) on the real line. The coordinate plane , or the \(x\)-\(y\) plane , is a geometric representation of \(\mathbb{R}^2\), the set of all pairs of real numbers. Each pair of real numbers \((a, b)\) is visualized as the point \((a, b)\) in the plane. Remark . Recall that \(\mathbb{R}^2 = \mathbb{R}\times\mathbb{R} = \{(x, y) | x, y \in \mathbb{R}\}\) is the Cartesian product of \(\mathbb{R}\) with i
Read more

Applications of multiple integrals

Image
We can use multiple integrals to compute areas and volumes of various shapes. Area of a planar region Definition (Area) Let \(D\) be a bounded closed region in \(\mathbb{R}^2\). \(D\) is said to have an area if the multiple integral of the constant function 1 over \(D\), \(\iint_Ddxdy\), exists. Its value is denoted by \(\mu(D)\): \[\mu(D) = \iint_Ddxdy.\] Example . Let us calculate the area of the disk \(D = \{(x,y)\mid x^2 + y^2 \leq a^2\}\). Using the polar coordinates, \(x = r\cos\theta, y = r\sin\theta\), \(dxdy = rdrd\theta\), and the ranges of \(r\) and \(\theta\) are \([0, a]\) and \([0, 2\pi]\), respectively. Thus, \[\begin{eqnarray*} \mu(D) &=& \iint_Ddxdy\\ &=&\int_0^a\left(\int_0^{2\pi}rd\theta\right)dr\\ &=&2\pi\int_0^a rdr\\ &=&2\pi\left[\frac{r^2}{2}\right]_0^a = \pi a^2. \end{eqnarray*}\] □ Volume of a solid figure Definition (Volume) Let \(V\) be a solid figure in the \((x,y,z)\) space \(\mathbb{R}^3\). \(V\) is sai
Read more