Concept Map
How the course's ideas depend on one another and where to start reading.
This page helps you find your way around the course. Read it once for an overview, then come back to the reading paths when a chapter uses a term you have not met. The ideas linking the two strands are set out on the Core page.
Questions to ask of a new term
When a new term appears, work out which of these questions it answers.
- What kind of object is it? A number, vector, matrix, polynomial, function, sequence, series, region or probability distribution. A space is a collection of objects with enough shared structure to be treated uniformly.
- What operations are available? Addition, scalar multiplication, composition, differentiation, integration, limits or applying a matrix. The operations decide which questions make sense. A vector space, for example, is defined by how its addition and scalar multiplication behave, not by drawing arrows.
- What problem is being solved? Most problems in the course do one of
the following:
- represent an object in coordinates;
- solve an equation or system;
- describe how a state evolves;
- approximate a hard object by a simpler one;
- optimise a quantity subject to conditions;
- accumulate a density over an interval or region.
- Which representation makes it simple? Complex numbers have Cartesian, polar and exponential forms. A vector can be drawn or given coordinates in a basis. A linear map can be written as a formula or a matrix. A planar region can be described in Cartesian or polar coordinates. Changing the representation leaves the object unchanged but changes which calculations are easy.
- What assumptions make the claim valid? Linearity, continuity, differentiability, positivity, independence, invertibility, distinct points and bounds on a region are not decoration. They are the conditions under which a conclusion is guaranteed.
- What kind of conclusion is it? An exact formula, existence without a formula, uniqueness, a qualitative description such as growth, decay or stability, an approximation with an error bound, or a numerical estimate whose accuracy depends on a step size or truncation. These differ in strength: knowing that a solution exists is not the same as being able to write it down.
The course in one picture
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│ Maps and change │ │ Accumulation │
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│ Modes and qualitative behaviour │ │ Mass, probability and averages │
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└─────────────────────────────────┘ - Objects: numbers, vectors, matrices, polynomials, functions, sequences and regions.
- Structure and representation: linear combinations, bases, coordinates, polar form and changes of variable.
- Maps and change: linear transformations, differentiation, recurrences, ODEs and matrix evolution.
- Modes and qualitative behaviour: eigenvalues, characteristic roots and stability.
- Approximation and optimisation: Taylor models, numerical methods, gradients, Hessians and constraints.
- Accumulation: integrals and double integrals, applied to mass, probability and averages.
The arrows show recurring dependencies, not a fixed teaching order. Taylor polynomials, for example, are at once vectors in a polynomial space, local approximations to functions and tools for numerical computation.
Reading paths
Algebra
- Vectors and scalars
- Linear combinations and span
- Linear independence
- Basis and dimension, then coordinates
- Linear transformations
- Kernel, image, rank and nullity
- Matrix representations
- Eigenvalues and diagonalisation
- Matrix dynamics and ODE systems
The chapters follow this order: complex numbers supplies the complex scalars, then vector spaces, linear transformations and eigenvalues.
Calculus
Everything starts from functions, variables and derivatives, then splits into three lines that can be read as each becomes relevant.
- Integration: antiderivatives and definite integrals, then integration techniques, then double integrals and changes of variables.
- Differential equations: initial data and solution families, first-order methods and models, second-order modes and forcing, existence, uniqueness and Lipschitz conditions, then numerical methods and error estimates.
- Local approximation: Taylor polynomials and their remainders, sequences and series, then gradients, Hessians and optimisation.
Notation
If the notation is the main obstacle, start with mathematical language. Return to it whenever two similar-looking terms start to blur together.