# 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   HELM Workbook See Written Resource
Complex arithmetic

Multiplying

Complex conjugates and dividing

Exponential form   HELM Workbook See Written Resource
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   HELM Workbook See Written Resource

#### Sequences and series

Sequences and series   Booklet See Written Resource
Infinite series   Booklet See Written Resource
Power series   Booklet See Written Resource
Binomial series   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
Newton Raphson Method Video Written Resource See Written Resource

#### Conic Sections

Conic Sections   HELM Booklet See Written Resource

#### Polar Coordinates

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

## Calculus

Video Written Resource Exercises

#### Differentiation

Differentiating sine and cosine from first principles   Workbook
Standard derivatives   Workbook
Chain rule Introduction to chain rule Workbook
Product rule   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

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   HELM Booklet See Written Resource
Cartesian components of vectors   HELM Booklet See Written Resource
Lines and planes   HELM Booklet See Written Resource
Scalar product   HELM Booklet See Written Resource
Vector product   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   HELM Booklet See Written Resource
Repeated eigenvalues and symmetric matrices   HELM Booklet See Written Resource
Numerical determination of eigenvalues and eigenvectors   HELM Booklet See Written Resource

#### Solving Systems of Equations

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

## Differential equations and transforms

Video Written Resource Exercises

#### Differential equations

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

Modelling with differential equations

Applications of differential equations

#### Fourier Transforms

Introduction Video HELM Workbook
Properties of Fourier transforms   HELM Workbook
Special Fourier transform pairs   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   HELM Workbook
Half-range series   HELM Workbook
The complex form   HELM Workbook
Applications of Fourier series   HELM Workbook

#### Laplace Transforms

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