# Next steps for STEM Subjects

These resources cover more advanced topics which may be useful to students on STEM courses.

On

## Algebra

Video Written Resource Exercises

#### Partial fractions

Partial fractions Introduction to partial fraction expansion Booklet See Written Resource

#### Complex Numbers

Complex arithmetic

Multiplying

Complex conjugates and dividing

Exponential form Video HELM Workbook See Written Resource
Complex arithmetic

Multiplying

Complex conjugates and dividing

Modulus-argument form

Finding modulus and argument

Writing in the modulus-argument form

Workbook See Written Resource
De Moivre's theorem and roots of complex numbers Video HELM Workbook See Written Resource

#### Sequences and series

Sequences and series Video Booklet See Written Resource
Infinite series Video Booklet See Written Resource
Power series Video Booklet See Written Resource
Binomial series Video Workbook See Written Resource

#### Binomial theorem

Binomial theorem Pascal's triangle and binomial expansion Workbook See Written Resource

#### Numerical Methods

Interval Bisection Video Webpage
Iteration Video
Newton Raphson Method Video Written Resource See Written Resource

#### Conic Sections

Conic Sections Video HELM Booklet See Written Resource

#### Polar Coordinates

Introduction Video HELM Booklet See Written Resource
Parametric Equations Video HELM Booklet See Written Resource

## Calculus

Video Written Resource Exercises

#### Differentiation

Differentiating sine and cosine from first principles Video Workbook
Standard derivatives Video Workbook
Chain rule Introduction to chain rule Workbook
Product rule Video Workbook Exercise
Quotient rule Introduction to the quotient rule   Exercise
Differentiation and trigonometry

Sin and cos

Tan and cot

Sec and cosec

Inverse trig functions

Differentiation with logs and exponentials Video

Differentiation by taking logs

Differentiating logs and exponentials

Higher-order derivatives Second derivatives
Stationary points

Maxima and minima

Points of inflection

Finding stationary points

HELM Workbook
Implicit differentiation

Implicit differentiation

Workbook
Parametric differentiation Introduction to parametric equations Workbook
Partial differentiation Introduction to partial derivatives

Partial derivatives Workbook

Stationary points Workbook

#### Integration

Methods for integration

Parts

Substitution

HELM workbook: Using a table of anti-derivatives

HELM workbook: Integration by parts

HELM workbook: Substitution

Integration with specific functions Using trig identities

HELM workbook: Integration leading to logs

HELM workbook: Trigonometric functions and integration

Multiple integrals

Double integrals

Volume integrals

HELM workbook: Introduction to surface integrals

HELM workbook: Multiple integrals over non-regular regions

HELM workbook: Volume integrals

#### Series

Maclaurin's and Taylor's series Taylor & Maclaurin polynomials intro Workbook

#### Numerical methods

Numerical differentiation Euler's method Workbook

## Vectors and matrices

Video Written Resource Exercises

#### Vectors

Introduction to Vectors Video HELM Booklet See Written Resource
Cartesian components of vectors Video HELM Booklet See Written Resource
Lines and planes Video HELM Booklet See Written Resource
Scalar product Video HELM Booklet See Written Resource
Vector product Video HELM Booklet See Written Resource

#### Matrices

Introduction to matrices Khan Academy Video HELM Booklet See Written Resource
Matrix multiplication

Multiplying by Scalars

Matrix Multiplication

HELM Booklet See Written Resource
Determinants Determinant of a 2x2 matrix HELM Booklet See Written Resource
The inverse of a matrix Khan Academy Video HELM Booklet See Written Resource
Eigenvectors and eigenvalues - Introduction Khan Academy Video HELM Booklet See Written Resource
Applications of eigenvalues and eigenvectors Video HELM Booklet See Written Resource
Repeated eigenvalues and symmetric matrices Video HELM Booklet See Written Resource
Numerical determination of eigenvalues and eigenvectors Video HELM Booklet See Written Resource

#### Solving Systems of Equations

Gaussian elimination Video HELM Booklet See Written Resource
Solution by Cramer's rule Video HELM Booklet See Written Resource
Solution by inverse matrix method Video HELM Booklet See Written Resource
Solution by Gaussian elimination Video HELM Booklet See Written Resource
LU decomposition Video HELM Booklet See Written Resource
Matrix norms Video HELM Booklet See Written Resource
Iterative methods for systems of equations Video HELM Booklet See Written Resource

## Differential equations and transforms

Video Written Resource Exercises

#### Differential equations

First-order differential equations Video HELM Workbook
Second-order differential equations Video HELM Workbook
Use of differential equations Video

Modelling with differential equations

Applications of differential equations

#### Fourier Transforms

Introduction Video HELM Workbook
Properties of Fourier transforms Video HELM Workbook
Special Fourier transform pairs Video HELM Workbook

#### Fourier Series

Periodic functions Video HELM Workbook
Representing periodic functions by Fourier series Video HELM Workbook
Even and odd functions Video HELM Workbook
Convergence Video HELM Workbook
Half-range series Video HELM Workbook
The complex form Video HELM Workbook
Applications of Fourier series Video HELM Workbook

#### Laplace Transforms

Causal functions Video HELM Workbook
The transform and its inverse Video HELM Workbook
Further Laplace transforms Video HELM Workbook
Solving differential equations Video HELM Workbook
The convolution theorem Video HELM Workbook
Transfer functions Video HELM Workbook

#### Partial Differential equations

Introduction Video HELM Workbook
Applications of PDEs   HELM Workbook
Solutions using separation of variables Video HELM Workbook
Solution using Fourier series Video HELM Workbook