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Ordering of infinite sets
Limits of infinite sequences
Properties of functions
Limits and continuous functions
Transcendental and real numbers
Univariate differentiation
Identifying and evaluating \(e\)
The sine and cosine functions, and identifying \(\pi\)
Polar coordinates
Power series, Taylor series and Maclaurin series
Matrix exponents and Taylor series of matrices
The Riemann integral, definite and indefinite integrals, and anti-derivatives
Integration by parts
The fundamental theorem of calculus
Lebesque integrals
The tangent function, and evaluating \(\pi\)
Other trigonometric functions
Fourier analysis
Generalising factorials: The gamma function
First-order Ordinary Differential Equations (ODEs)
Second-order Ordinary Differential Equations (ODEs)
Univariate optimisation
Multivariate functions
Multivariate differentiation of scalar fields - partial differentiation, del and grad
Total differentiation of scalar fields
Directional derivative of scalar fields
Multivariate integration of scalar fields
Generalising the binomial coefficient formula: The beta function
Multivariate optimisation of scalar fields
Multivariate differentiation of vector fields, including Jacobians, scalar potential, conservative vector fields, divergence, Solenoidal vector fields, Laplace operator, curl, hodge stars and hodge duals
Multivariate integration of vector fields
Partial Differential Equations (PDEs)
Variational calculus/functionals
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