Introduction

Physics describes the physical world using precise mathematical language, and this chapter introduces two mathematical tools that you will use throughout your study of physics: vector analysis and an elementary introduction to differential and integral calculus.

In earlier study you have met physical quantities such as length, mass, and time, which are completely described once you state a number and a unit. But many physical quantities you will meet in mechanics and beyond — displacement, velocity, force, acceleration, momentum — cannot be fully described by a magnitude alone; they also need a direction. Vector analysis gives you the systematic language and rules (addition, subtraction, resolution into components, and two distinct kinds of multiplication) needed to handle such quantities correctly.

Separately, many physical situations involve quantities that change continuously — the position of a moving body changing with time, or the velocity of a body changing under a force. Calculus is the branch of mathematics built specifically to handle continuous change: differential calculus lets you find the instantaneous rate at which one quantity changes with another (for example, velocity as the rate of change of position), while integral calculus lets you find the cumulative effect of a continuously changing quantity (for example, the total area under a graph, or the distance covered from a varying velocity). You will study both branches in full mathematical depth in Class XII mathematics; here you learn only as much as is needed to follow the physics developed in the rest of this book.