University of Oxford Physics Year 1 - Calculus, Waves and Normal Modes (CP4) Complete Lecture Notes & Revision Guide

Comprehensive handwritten lecture notes for the University of Oxford Year 1 Physics course "Calculus, Waves and Normal Modes (CP4)". These notes provide a structured and detailed treatment of the material covered throughout the course, including derivations, worked examples, key formulas, diagrams, and explanations designed to support both understanding and revision. Topics covered include: • Differentiation from first principles, differentiation rules, implicit differentiation, parametric differentiation, logarithmic differentiation, and higher-order derivatives • Integration from first principles, the Fundamental Theorem of Calculus, reduction formulae, integration by parts, change of variables, and properties of definite integrals • Applications of integration including volumes of revolution, surface areas, arc length, and physical applications of integrals • Taylor and Maclaurin series, polynomial approximations, series manipulations, and local approximations of functions • Functions of several variables, partial derivatives, higher-order derivatives, total differentials, chain rules, and variable transformations • Taylor expansions of multivariable functions, small changes and approximations, maxima, minima, saddle points, and optimisation problems • Reciprocity relations, cyclic relations, and coordinate transformations in multivariable calculus • Double integrals, Jacobians, and integration in Cartesian, polar, cylindrical, and spherical coordinate systems • Line integrals, surface integrals, volume integrals, scalar and vector fields, and conservative fields • Gradient, divergence, curl, and applications of Gauss' and Stokes' theorems • Probability theory, probability distributions, and mathematical tools used in physical applications • Coupled oscillations, normal frequencies, normal modes, and energy in oscillatory systems • Derivation and solution of the one-dimensional wave equation, travelling waves, standing waves, reflection, transmission, impedance, beats, wave packets, phase velocity, group velocity, and separation of variables The document consists of carefully organised handwritten notes taken during the course and is suitable for lecture review, tutorial preparation, revision, and exam preparation. Ideal for Oxford Physics students and anyone studying introductory university-level calculus, vector calculus, waves, and mathematical methods.