# Next steps for STEM Subjects

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

## Algebra

Video | Written Resource | Exercises | |
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## Partial fractions |
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Partial fractions | Introduction to partial fraction expansion | Booklet | See Written Resource |

## Complex Numbers |
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Complex arithmetic | |||

Exponential form | HELM Workbook | See Written Resource | |

Complex arithmetic | |||

Exponential form | HELM Workbook | See Written Resource | |

Modulus-argument form | Workbook | See Written Resource | |

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

## Sequences and series |
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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 |
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Binomial theorem | Pascal's triangle and binomial expansion | Workbook | See Written Resource |

## Numerical Methods |
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Interval Bisection | Video | Webpage | |

Iteration | |||

Newton Raphson Method | Video | Written Resource | See Written Resource |

## Conic Sections |
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Conic Sections | HELM Booklet | See Written Resource | |

## Polar Coordinates |
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Introduction | HELM Booklet | See Written Resource | |

Parametric Equations | HELM Booklet | See Written Resource |

## Calculus

Video | Written Resource | Exercises | |
---|---|---|---|

## Differentiation |
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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 | |||

Differentiation with logs and exponentials | |||

Higher-order derivatives | Second derivatives | ||

Stationary points | HELM Workbook | ||

Implicit differentiation | Workbook | ||

Parametric differentiation | Introduction to parametric equations | Workbook | |

Partial differentiation | Introduction to partial derivatives | ||

## Integration |
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Methods for integration |
HELM workbook: Using a table of anti-derivatives |
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Integration with specific functions | Using trig identities | ||

Multiple integrals |
HELM workbook: Introduction to surface integrals |
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## Series |
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Maclaurin's and Taylor's series | Taylor & Maclaurin polynomials intro | Workbook | |

For Fourier Series click here | |||

## Numerical methods |
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Numerical differentiation | Euler's method | Workbook |

## Vectors and matrices

Video | Written Resource | Exercises | |
---|---|---|---|

## Vectors |
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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 |
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Introduction to matrices | Khan Academy Video | HELM Booklet | See Written Resource |

Matrix multiplication | HELM Booklet | See Written Resource | |

Matrix addition and subtraction | Khan Academy Video | ||

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 |
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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 |
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First-order differential equations | HELM Workbook | ||

Second-order differential equations | HELM Workbook | ||

Use of differential equations | |||

## Fourier Transforms |
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Introduction | Video | HELM Workbook | |

Properties of Fourier transforms | HELM Workbook | ||

Special Fourier transform pairs | HELM Workbook | ||

## Fourier Series |
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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 |
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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 |
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Introduction | Video | HELM Workbook | |

Applications of PDEs | HELM Workbook | ||

Solutions using separation of variables | Video | HELM Workbook | |

Solution using Fourier series | Video | HELM Workbook |

## Book a 1:1 appointment or workshop

Would you like to explore a maths or stats topic in greater depth? Why not book a 1:1 with an advisor or a workshop (current students only).