Education
An Overview of Mathematics in Classical Mechanics
Classical mechanics, developed primarily through the work of Sir Isaac Newton and subsequent physicists such as Euler, Lagrange, and Hamilton, provides the foundation for understanding the motion of physical systems under the influence of forces.
Derivation and Examples of Reduction Formulas
In this section, we will derive and provide examples for various reduction formulas used in calculus. Each formula allows us to reduce the complexity of integrals involving different types of functions, such as exponential, trigonometric, inverse trigonometric, hyperbolic, logarithmic, and algebraic functions.
A Comprehensive Guide to Series in Real Analysis
In real analysis, series play an important role in understanding the convergence properties of sequences and functions. A series is essentially the sum of the terms of a sequence, and the study of series provides the foundation for advanced concepts such as Taylor series, Fourier series, and more.
A Comprehensive Guide to Vectors, Kernel, Green’s Theorem, Bulk Modulus, Gauss’s Theorem with Examples
In the study of mathematics and physics, concepts like vectors, kernel, Green’s theorem, bulk modulus, and Gauss’s theorem play vital roles in a variety of fields, ranging from fluid dynamics to material science.
Solving Ordinary Differential Equations (ODEs) – 1st Order
Ordinary Differential Equations (ODEs) are equations involving functions and their derivatives. First-order ODEs involve the first derivative of the unknown function and are crucial in modeling phenomena such as population dynamics, heat conduction, motion under forces, and electrical circuits.
Understanding the Wronskian, Eigenvalues, and Eigenvectors
In the realm of linear algebra and differential equations, Wronskian, eigenvalues, and eigenvectors play a crucial role in solving systems of linear equations, studying the behavior of differential equations, and analyzing the stability of systems. In this blog, we will dive deep into each of these concepts, provide formal definitions, and illustrate them with practical…
Order of Partial Differential Equations (PDEs)
In the study of partial differential equations (PDEs), one of the key characteristics used to classify and understand them is the **order** of the equation. The order of a PDE refers to the highest derivative of the unknown function in the equation. Understanding the order is crucial for identifying the complexity of the equation and…
Classification of Partial Differential Equations (PDEs)
Partial Differential Equations (PDEs) are equations involving functions of several variables and their partial derivatives. They arise in various fields such as physics, engineering, economics, and biology. To understand and solve PDEs, it is crucial to first classify them. The classification helps to determine the nature of the equation and the appropriate methods for finding…
Solving Differential Equations Using the Laplace Transform: A Step-by-Step Guide
Abstract: Differential equations are fundamental in mathematics, physics, and engineering, as they describe how quantities change over time. One powerful method for solving ordinary differential equations (ODEs) is the Laplace transform. This blog will introduce the Laplace transform and demonstrate its application in solving a simple first-order ODE with a detailed example. Table of Contents…
Understanding Fourier Series: Breaking Down Functions into Sines and Cosines
Fourier series is a powerful concept in mathematics and engineering that allows us to express periodic functions (functions that repeat their values in regular intervals) as the sum of simple sine and cosine waves.
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