# Mathematics Support

## Contents

- 1 Arithmetic
- 2 Co-ordinate Systems
- 3 Basic Algebra
- 4 Functions
- 5 Sets
- 6 Trigonometry
- 7 Differentiation
- 8 Further Techniques of Differentiation I
- 9 Optimisation Problems
- 10 Further Techniques of Differentiation II
- 11 Multivariate Calculus
- 12 Integration
- 13 Further Algebra
- 14 Maclaurin and Taylor Series
- 15 Further Integration
- 16 Numerical Methods
- 17 Differential Equations
- 18 Vectors
- 19 Complex Numbers
- 20 Matrices
- 21 Double Integrals
- 22 Integral Transformations

## Arithmetic

## Co-ordinate Systems

## Basic Algebra

In this section you will be investigating how to manipulate algebraic equations, and how to solve linear equations. We will then extend this to solving linear simultaneous equations. Finally we will be investigating quadratic equations.

## Functions

In this section you will be investigating the definition of a mapping, and the various types of mappings. You will then familiarise yourself with the definition of a function before you finally move on to the definition of the domain and range.

## Sets

In this section you will be investigating various properties of sets. Beginning with the definition of a set you will move on to some examples of sets from real life to those used extensively in mathematics. We will then move on to the notation that we use in sets, and finally the basic algebra of sets.

## Trigonometry

## Differentiation

In this section you will get a feel for the basic idea behind differentiation, and what is meant when we discuss ideas such as rate of change of a function. We will then apply the basic ideas of differentiation to several different functions including monomials/polynomials, exponentials, trigonometric and logarithmic functions.

## Further Techniques of Differentiation I

In this section we will take the basic ideas of differentiation that we learnt in the previous section and extend them to a function of a function, products of functions and quotients of functions.

## Optimisation Problems

For this section we will use the ideas from the second derivative to optimise certain physical/real life problems. We will introduce a second method to solve optimisation problems under certain constraints using Lagrange Multipliers.

## Further Techniques of Differentiation II

## Multivariate Calculus

In this section will extend the ideas of uni-variate calculus to that of functions of more than one variable, what we call multivariate calculus.

## Integration

In this section we will introduce what integration is, and the physical meaning of integration. We will then investigate how to integrate various functions, including monomials/polynomials, exponentials and trigonometric functions.

## Further Algebra

In this section you will investigate further algebraic techniques such as partial fractions and transformations of functions.

## Maclaurin and Taylor Series

In this section we will derive what is meant by a Maclaurin and Taylor series. We then use them to help solve situations where we can apply approximations.

## Further Integration

For this section we will be extending the ideas of integration which were discussed in the previous section to those of products of functions, and functions of a function, integration by parts and integration via substitution.

## Numerical Methods

## Differential Equations

In this section we will learn the various different methods for solving first and second order differential equations with constant/non-constant coefficients.

## Vectors

In this section we will introduce what is meant by a vector and compare vector quantities to scalar quantities. We will then introduce vector algebra, including 2D and 3D cartesian vectors. We will finish this section by considering the dot product, and the cross product.

## Complex Numbers

## Matrices

In this section we will begin with the basics of matrices, including the definition of a matrix and matrix algebra. We will then move on to solving systems of linear equations using Gaussian Elimination. We will then use Gaussian elimination to find matrix inverses and introduce elementary matrices as an alternative approach. Finally we will develop the method of LU decomposition as an alternative method for solving systems of linear equations.

## Double Integrals

## Integral Transformations

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