👉 To explain the full math behind a topic, we start with defining the core concepts and principles involved. For instance, in calculus, we delve into limits, derivatives, and integrals to understand rates of change and accumulation. Limits describe how functions behave as input values approach a specific point, laying the groundwork for derivatives which measure instantaneous rates of change. Derivatives are calculated using the limit definition, involving the difference quotient: f'(x) = lim(h → 0) [f(x + h) - f(x)]/h. Integrals, conversely, represent accumulation of a function over an interval, calculated via the definite integral: ∫f(x)dx = F(x) + C, where F(x) is the antiderivative of f(x). In linear algebra, we explore vector spaces, linear transformations, and matrices, utilizing concepts like eigenvalues, eigenvectors, and determinants to solve systems of equations and analyze transformations. Differential equations, which describe how quantities change over time or space, involve techniques like separation of variables, integrating factors, and characteristic equations to find solutions. Probability theory, meanwhile, employs random variables, probability distributions (like the normal and binomial), and the law of large numbers to model uncertainty and make predictions. Each branch builds upon foundational principles, offering a comprehensive framework for problem-solving across mathematics and its applications.
