A differential equation is a mathematical equation that relates a function with its derivatives, and is used to describe various phenomena in physics, engineering, biology, and other disciplines where the rate of change is essential.

Double integrals are an extension of integration to functions of two variables and are used to compute volumes or accumulated quantities over two-dimensional regions.

The Laplacian is a second-order differential operator widely used in mathematics, physics, and engineering to study phenomena like heat conduction, fluid flow, and electromagnetism.

Separation of variables is a technique used to solve partial differential equations by separating the variables into multiple independent single-variable functions, often used for linear PDEs.

Simpson’s Rule is a numerical integration technique that approximates the definite integral of a function by fitting a series of parabolas over equally spaced subintervals.

Gaussian quadrature is a numerical integration technique used to approximate definite integrals of functions, providing more accurate results than other methods, especially for smooth functions, with various types such as Legendre, Chebyshev, and Hermite quadrature.

Second derivatives represent the rate of change of the first derivative or the acceleration of the original function, essential in calculus for analyzing curvature and concavity, and used in various applications such as optimization and partial differential equations.

Power series are infinite series used to represent a wide variety of functions, with applications in calculus, complex analysis, and numerous fields like physics, engineering, and finance.

Taylor’s Formula approximates a function by an infinite sum of its derivatives evaluated at a single point, serving as the foundation for Taylor series and Maclaurin series.