Yao Zhang - Analysis

Analysis is a branch of mathematics that studies limits, continuity, derivatives, integrals, and infinite series. It provides essential tools for understanding change and motion, making it vital in fields like physics, engineering, and economics. Key concepts include real numbers, sequences, and functions, which enhance our understanding of mathematical behavior.

A significant aspect of analysis is the rigorous examination of limits, leading to formal definitions of continuity and differentiability. The derivative indicates the rate of change of a function, while the integral focuses on accumulation. Differential equations, which describe relationships between functions and their derivatives, are crucial for modeling real-world phenomena, such as population growth and fluid dynamics. The Fundamental Theorem of Calculus connects differentiation and integration, highlighting their interrelated nature.

Overall, analysis underpins modern mathematics, offering tools for rigorous reasoning and problem-solving, with applications in differential equations that allow for the modeling of complex systems in science and engineering.