Mathematical Foundations

Mathematics is the shared language of every technical domain in this vault. These notes are not abstract exercises — every concept here connects directly to power systems analysis, control theory, machine learning, or signal processing.

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Core Topics

Calculus

Limits, derivatives, integrals, multivariable calculus, vector calculus. The language of rates of change and accumulation.

$\frac{d}{dt}{\int }_a^{g(t)}f(x)dx=f(g(t))\cdot g^{\prime }(t)$

Linear Algebra

Vectors, matrices, eigenvalues, eigenvectors, decompositions. The language of systems and transformations.

$A\mathbf{x}=\lambda \mathbf{x}$

Differential Equations

ODEs, PDEs, Laplace transforms, state-space methods. The language of dynamic systems.

$\mathcal{L}\{f(t)\}={\int }_0^{\infty }f(t)e^{-st}dt$

Probability & Statistics

Probability spaces, distributions, expectation, hypothesis testing, Bayesian inference.

Discrete Mathematics

Logic, set theory, combinatorics, graph theory, number theory. The language of algorithms and computation.

Complex Analysis

Complex numbers, phasors, Fourier analysis, frequency domain methods. Essential for AC power and signals.

$e^{j\theta }=\cos \theta +j\sin \theta$

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Key Questions These Notes Answer

• How do I model a dynamic system mathematically?
• What does an eigenvalue mean physically in a power system?
• How does a Fourier transform connect time and frequency domains?
• How do I quantify uncertainty in an engineering measurement?

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Prerequisites

High school algebra and trigonometry.

Connects To

• Control Systems — differential equations, Laplace transforms
• Power Systems — complex analysis, phasors, linear algebra
• Data Science & AI — linear algebra, probability, statistics
• FE Exam — mathematics is the largest section