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Comprehensive derivation of Fourier Series coefficients, covering orthogonality relations, integration techniques, and uniform convergence. Includes practical examples and a free worksheet for hands-on practice.
Comprehensive explanation of Taylor's Theorem, including derivation and practical examples. Covers polynomial approximation of functions, Cauchy's Mean Value Theorem, and power series expansion, with interactive exercises and additional resources.
Learn to calculate matrix inverses using Elementary Row Operations (ERO's) with clear explanations and worked examples for 2x2 and 4x4 matrices, including the case of non-invertible matrices.
Comprehensive tutorial on solving linear equations using matrices and elementary row operations, with worked examples and practice resources for self-study.
Explore a challenging cube coloring problem and learn how to solve it using mathematical reasoning and combinatorial analysis.
Learn to solve homogeneous first order differential equations with Oxford mathematician Dr Tom Crawford. Includes worked examples, partial fractions, implicit differentiation, and practice exercises.
Derive the quadratic formula and apply it to real-world scenarios in football and archery. Learn 'completing the square' method and solve problems in penalty kicks and projectile motion.
Learn to solve first order linear differential equations using integrating factors. Understand the technique's derivation, application, and practice with a provided worksheet and example problems.
Learn to classify 2D critical points using the discriminant method. Explore maximum, minimum, and saddle points with visualizations and step-by-step derivations for multivariable functions.
Explore the fascinating world of 3D fractals, from natural examples to complex mathematical constructs like Mandelbulbs, with insights into fractal dimensions and generation techniques.
Explore complex mathematics behind the Multiverse Library, focusing on branch cuts and multifunctions. Learn about holomorphic branches of complex logarithms and square root functions in this engaging mathematical journey.
Comprehensive explanation of direct sum of vector spaces with worked examples. Learn formal definitions, alternative representations, and how to verify direct sums through intersection and spanning checks.
Explore the dimension formula for vector spaces through worked examples and a complete proof. Learn to calculate dimensions of subspaces and their intersections in R^4.
Explore key concepts in linear algebra: basis, spanning, and linear independence. Learn through intuitive examples and formal definitions, with worked problems in vector spaces and Cartesian planes.
Comprehensive explanation of gradient vector and divergence in multivariable calculus, including formal definitions, visualizations, examples, and connections to other concepts. Includes free worksheet and app recommendations.
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