Advanced Mathematics · University & competition

University Year 1 Mathematics

An extra-deep first-year BSc mathematics core with 15 units, 75 rigorous lessons and adaptive one-question practice across pure, applied, statistical and computational mathematics.

BSc Mathematics core · Year 1

15 topics

75 lessons

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What you will be able to do

Write rigorous proofs from definitions
Use single and multivariable calculus fluently
Solve linear systems and reason in vector spaces
Build probability, statistics and differential-equation models
Validate mathematical and computational conclusions

Course contents

Every topic, lesson by lesson.

Term 1 · Unit 1

Proof and mathematical language

Logic, quantifiers, sets, functions and rigorous proof construction.

Term 1 · Unit 2

Calculus foundations

Limits, continuity and derivatives built from precise local reasoning.

Term 1 · Unit 3

Integral calculus

Definite integration, the fundamental theorem and systematic techniques.

Term 1 · Unit 4

Linear systems and matrices

Row reduction, matrix algebra, determinants and invertibility.

Term 1 · Unit 5

Vector spaces

Subspaces, span, independence, bases and dimension.

Term 2 · Unit 6

Sequences and series

Convergence, monotonicity, infinite series and power series.

Term 2 · Unit 7

Multivariable calculus

Geometry and differentiation of scalar functions of several variables.

Term 2 · Unit 8

Probability

Axioms, conditioning, random variables and expectation.

Term 2 · Unit 9

Statistical foundations

Sampling, estimation, uncertainty and introductory inference.

Term 2 · Unit 10

Discrete mathematics

Counting, recurrences, graph structure and finite algorithms.

Term 3 · Unit 11

Differential equations

First- and second-order differential equations and qualitative models.

Term 3 · Unit 12

Numerical methods

Approximation, error, roots, interpolation and quadrature.

Term 3 · Unit 13

Mathematical mechanics

Vector motion, forces, energy and momentum in mathematical models.

Term 3 · Unit 14

Mathematical computing

Algorithms, numerical experiments and reproducible mathematical reasoning.

Term 3 · Unit 15

Year 1 mathematical synthesis

Connect proof, pure mathematics, statistics and modelling.

Concepts to practise again

Built from your saved practice attempts in this course, weakest question types first.

Complete a practice set in any lesson to see which question types to practise again.

Curriculum alignment

This course satisfies the following education systems.

British Curriculum