Learn about the definition of gravitational field vectors and gravitational potential according to Newton's law of universal gravitation, and examine two important related examples: the shell theorem and galactic rotation curves.
Explore the method of undetermined coefficients, a powerful technique for solving specific nonhomogeneous linear ODEs with constant coefficients, widely used in engineering for models like vibrating systems and RLC circuits.
Explore the structure of the general solution for second-order nonhomogeneous linear ODEs in relation to their homogeneous counterparts. This post proves the existence of a general solution and the non-existence of singular solutions.
Explore the existence and uniqueness of solutions for second-order homogeneous linear ODEs with continuous variable coefficients. Learn to use the Wronskian to test for linear independence and see why these equations always have a general solution that encompasses all possible solutions.
Explore the different forms of the general solution to the Euler-Cauchy equation based on the sign of the discriminant of its auxiliary equation.
A comprehensive examination of various methods for determining whether a series converges or diverges.
We examine fundamental concepts of calculus such as the definition of sequences and series, convergence and divergence of sequences, convergence and divergence of series, and the definition of e, the base of natural logarithm.
We explore Newton's laws of motion, the meaning of these three laws, and the definitions of inertial mass and gravitational mass, as well as the principle of equivalence, which holds significant importance not only in classical mechanics but also in the later theory of general relativity.
Explore how the general solution of a second-order homogeneous linear ODE with constant coefficients changes based on the roots of its characteristic equation.
This post introduces the process of implementing multi-language support on a Jekyll blog based on the 'jekyll-theme-chirpy' by applying the Polyglot plugin. This is the third post in the series, covering the identification and resolution of errors that occur when applying Polyglot to the Chirpy theme.