Foundations of Physics: Learning Newton's Calculus

This book is a mathematical preparation for students about to begin calculus-based mechanics. Its aim is to bridge the gap from basic algebra and trigonometry to the calculus and linear-algebra tools routinely used in first-year physics. Differentiation is developed from tangents and limits to the standard rules; integration is introduced as the inverse of differentiation, grounded in area and Riemann sums. Trigonometry (radians, the unit circle, small-angle approximations, and core identities), vectors and their algebra, basic matrices and coordinate transformations, rotations, polar coordinates, and introductory differential equations (e.g., exponential growth and decay) are presented where they naturally arise in simple mechanics scenarios.The scope is deliberately mathematical: physical examples — constant velocity and acceleration, circular motion, and coordinate changes — serve to illuminate the mathematics rather than to build a full mechanics curriculum. Throughout, the text emphasizes clear reasoning, multiple representations, and tidy derivations supported by worked examples. Exercises, including 'Brain Gym' problem sets, promote active engagement and steady skill-building, while brief historical notes and an enthusiastic, direct style invite students to view physics as a living discipline.Designed as a bridge or companion for calculus-based mechanics, this book assumes only algebra and basic trigonometry, and prepares students for subsequent study in physics, engineering, and applied mathematics by cultivating both computational fluency and conceptual clarity.