Foundations of Physics: Vectors, Calculus, and More

1. Introduction to Physics • Introduction to Vectors • Introduction to Calculus(微积分) Chapter 0 Preface

2. Chapter 0 Preface • Introduction to Physics (See 动画库\力学夹\绪论.exe) 1) Objects studied in physics 2) Methodology for studying physics 3) Some other key points

3. Chapter 0 Preface • Introduction to Vectors A scalar is a simple physical quantity that does not depend on direction. mass, temperature, volume, work… A vectoris a concept characterized by a magnitude and a direction. force, displacement, velocity…

4. Chapter 0 Preface 1) Representation of vectors (See 动画库\力学夹\0-4矢量运算.exe) 2) Addition and subtraction of vectors (See 动画库\力学夹\0-4矢量运算.exe) 3) Dot and cross products

5. 3.1) Dot product: No problem， if ? ? Chapter 0 Preface

6. Prove it? Chapter 0 Preface

7. Scalar triple product: Chapter 0 Preface 3.2) Cross product: The length of   ×  can be interpreted as the area of the parallelogram having A and B as sides. is a unit vector perpendicular to both and . , , and also becomes a right handed system.

8. Chapter 0 Preface

9. Note that this statement can be true even if       or ƒ(x) is not defined at c. Chapter 0 Preface • Introduction to Calculus(微积分) 1) Limit of a function ƒ(x) can be made to be as close to L as desired by making x sufficiently close to c. “The limit of ƒ of x, as x approaches c, is L." Example:

10. s s t1 t2 t1 t2 t t ? Chapter 0 Preface 2) Derivative of a function(函数的导数) • Motion with constant • velocity • Motion with changing • speed

11. Chapter 0 Preface • Motion with changing speed How to find the instantaneous speed at t1? Derivative of s

12. f(x) A’ tangent A x1 x2 x Chapter 0 Preface For general function, its derivative is defined as: The meaning of derivative of a function:

13. is infinitesimal. How big is an infinitesimal?...

14. Chapter 0 Preface

15. Chapter 0 Preface Example: Some basic formulae:

16. Chapter 0 Preface Some basic rules: ,C is a const. For a vector:

17. Chapter 0 Preface 3) Differential of a function (函数的微分) If f(x) has its derivative at point x, then f ’(x)dx is its differential at that point. Differential of the function Differential of the variable So f ’(x) is also called differential quotient (微商)

18. Chapter 0 Preface Operation rule is the same as that for derivative: ,C is a const. One application of differential

19. Chapter 0 Preface Example: Following approximate formulae often used in physics ( ) :

20. v v S S t0 t0 t t 0 0 Chapter 0 Preface 4) Integrals(积分) • Motion with constant • velocity • Motion with changing • speed How to find S?

21. Chapter 0 Preface v … … t i t0 0 In general, the integral from a to b of f(x) with respect to xis expressed as: definite integral indefinite integral

22. Chapter 0 Preface How to find an integral of a function? If function f(x) is continuous on the interval [a, b] and if on the interval (a, b), then

23. Chapter 0 Preface Example: Basic integral formulae: k,C: const.