1. LESSON 1 Mathematics for Physics
Mr. Komsilp Kotmool (Aj Tae)
Department of Physics, MWIT
Web site : .ac.th/~tae_mwit

2. Warm Up the farmer needs 3 hens to produce 12 eggs in 6 days.
A single chicken farmer has figured out that a hen and a half can lay an egg and a half in a day and a half. How many hens does the farmer need to produce one dozen eggs in six days?
A single chicken farmer also has some cows for a total of 30 animals, and all animals in farm have 74 legs in all. How many chickens does the farmer have?
the farmer needs 3 hens to
produce 12 eggs in 6 days.

3. CONTENTS Why do we use Mathematics in Physics?
What is Mathematics in this course?
Functions
Limits and Continuity of function
Fundamental Derivative
Fundamental Integration
Approximation methods

4. Why do we use Mathematics in Physics (and other subjects)?
ข้อมูลเชิงวิทยาศาสตร์ (ฟิสิกส์)
Qualitative Physics – เชิงคุณภาพ เป็นรูบแบบการศึกษาที่บ่งบอกถึงความรู้สึก มีหรือไม่มี เช่น บางส่วนของวิชา Quantum Physics
Quantitative Physics – เชิงปริมาณ เป็นรูปแบบการบ่งชี้เชิงข้อมูที่แม่นตรง มาก-น้อยแค่ไหน สามารถนำไปประยุกต์ใช้ในกิจกรรมของมนุษย์ได้ย่างมีประสิทธิภาพ (ส่วนใหญ่ของฟิสิกส์)

5. What is Mathematics in this course?
REVIEW: basic problem
Displacement S = ? = v x t NOT AT ALL
condition: v must be constant !!
But in reality, v is not constant !!!
General definition
Better or
CALCULUS or

6. Functions What is FUNCTION? REVIEW : some basic equations
y = Ax+B linear
y = Ax2+Bx+C parabola
y = sin(x) trigonometric
y = circular
y is the function of x for the 1-3 equation, but the 4th is not!!
Rewrite: y is the function of x to be y(x) [y of x]

7. Functions What is FUNCTION? REVIEW again: For physical view
The motion of a car is expressed with S = 5t-5t2 equation.
Displacement (S) depends on time (t)
We can say that S is the function of t, and can be rewritten:
S(t) = 5t-5t2
Note: We usually see f(x) and g(x) in many text books.

8. Functions Mathematical definitions
1). If variable y depends on a variable x in such a way that each value of x determines exactly one value of y, then we say that y is a function of x.
2). A function f is a rule that associates a unique output with each input. If a input denoted by x, then the output is denoted by f(x) (read “f of x”)

9. Exercise
Consider variable y whether it is the function or not and find the value of y at x=2
1). y – x2 = 5
2). 4y2 + 9x2 = 36
3). y = 0
4). ycsc(x) = 1
5). y2 +2y –x =0

10. Limits and Continuity of function
What is limits?
Why do we use limits?
When do we use limits?
?????

11. Limits and Continuity of function
Newton’s Law of Universal Gravitation
We can not calculate force at r → ∞.
How do we solve this problem??
But we can find the close value!!!!!!
at r → ∞.

12. Limits and Continuity of function
Electric field at arbitrary from solid sphere
E = ? At r = a

13. Limits and Continuity of function
For mathematical equation
f(x=0) = ?
We can not find f(x) at x = 0
But we can find its close value of f(x) at x → 0
Therefore, f(x) → 2 at x → 0
“the limit of f(x) as x approaches 0 is 2”

14. Limits and Continuity of function
How about at x → 0?
-1 ; x<0
1 ; x>0
Limit from the left
Limit from the right
Not exist !!
For discontinuity

15. Limits and Continuity of function
THEOREMS. Let a and k be real numbers, and suppose that
and 1 2 3 4 5
and

16. Limits and Continuity of function
A function f(x) is said to be continuous at x = c provided the following condition are satisfied :
1). f(c) is defined
2). exists
3).

17. History of Calculus Sir Isaac Newton (1642-1727)
Gottfried Wilhelm Leibniz ( )

18. Fundamental Derivative
In reality, the world phenomena involve changing quantities:
the speed of objects (rocket, vehicles, balls, etc.)
the number of bacteria in a culture
the shock intensity of an earthquake
the voltage of an electrical signal
It is very easy for the ideal situations {many exercises in your text books}
For example: constant of velocity
S = v x t
rate of change of S is constant v = ΔS/Δt
Consider: can we have this situation in the real world?

19. Fundamental Derivative
There are many conditions in nature affecting to complicate phenomena and equations!!!

20. Fundamental Derivative
Consider: equation of motion (free fall)
S(t) = ut – ½(g)t2
Velocity does not be constant with time !!!
How do we define velocity from t1 to t2 ?
Average velocity ?

21. Fundamental Derivative
Slopes and rate of change
Take t2 close to t1
S(t)
S(t) t t1 t2
S(t1)
S(t2)
Close to tangent line t1 t2
S(t1)
S(t2) t
vav can not be exactly represented this motion!
more exactly !!!
Slope at t = t1 is tangent line

22. Fundamental Derivative
Take t2 close to t1 that t2-t1 = Δt → 0
vint → v(t) instantaneous velocity
Mathematical notation
(read “dee S by dee t”)

23. Fundamental Derivative
Exercises: Find the derivative of y(x) and its value at x=2
y(x) = x
y(x) = x2
y(x) = sin(x)

24. Fundamental Derivative
FORMULARS: If f(x) and g(x) are the function of x and c is any real number 1. 2. 3. 4. 5. 6.

25. Fundamental Derivative
What is the derivative of these complicate functions?
Case 1) y(x) = sin(x2)
Case 2) y(x) = xcos(x)
Case 3) y(x) = tan(x)
Chain rule:

26. Fundamental Derivative
FORMULARS : If u is a function of x {u(x)}, and c is any real number
Frequency use in Physics 6. 7. 1. 2. 3. 4. 5.
g(x) ≠ 0

27. Additional applications of Derivative
Maximum and Minimum problems
Ex : An open box is composed of a 16-inch by 30-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares to be obtained a box with the largest volume?
16 in
30 in x

28. Fundamental Integration
What is the integration (calculus)?
Why do we use the integration (calculus)?
When do we use the integration (calculus)?

29. Fundamental Integration
Recall: If we know S(t) of a object  we can find v(t) of its and say v(t) is the derivative of S(t) !!
Reverse problem : If we know v(t) of a object  Can/How do we find S(t) of its ?
We can say that S(t) is the antiderivative of v(t) !!
What is the antiderivative?

30. Fundamental Integration
Consider graphs of constant and linear velocity with time (v(t)-t)
v(t) t v0 v
S = v0t
S = (1/2)(v0+v)t
S(t) was represented by area under function of v(t) !!!

31. Fundamental Integration
How about the complicate function of v(t)?
v(t) t v
What about S(t) of this curve?
S(t) also was represented by area under of this curve!
Next problem….
How do we find this area?

32. Fundamental Integration
Method of Exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.
Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater and greater number of sides.
disadvantage: do not proper with asymmetric shape

33. Fundamental Integration
v(t) t v
For smoother area t*
v(t*)
Therefore, we get Δt
Read “S(t) equal integral of v(t) dee t”

34. Fundamental Integration
Mathematical Method
Note : f(x) is the derivative of F(x), but in the other hand, F(x) is a antiderivative of f(x).
For Polynomial Function
F(x) = x2
f(x) = 2x
Constants depend on conditions !!
F(x) = x2 + 5
f(x) = 2x
F(x) = x2 + 10
f(x) = 2x

35. Fundamental Integration
Recall:
Set n=n-1
Therefore, we get
For the flexible form
When C is the arbitrary constant.

36. Fundamental Integration
Exercises: Find the antiderivative of the following y(x).
y(x) = 3x3 + 4x2 + 5
y(x) = 0
y(x) = sin(x)
y(x) = xcos(x2)
y(x) = tan(x)
Integration by Substitution

37. Fundamental Integration
FORMULARS: If f(x) and g(x) are the function of x and c is any real number 1. 2. 3. 4. 5. 6. 7. 8. 9.
10.
All of these are call the indefinite integral

38. Fundamental Integration
The definite integral
Recall:
We get the general solution of S(t) from the indefinite integral.
Problem: How do we get displacement (S) in the
interval [a,b] of time?
v(t) t
a b
a is the lower limit
b is the upper limit
this is the definite integral !

39. Fundamental Integration
Definition:
(a) If a is in the domain of f(x), we define
(b) If f(x) is integrable on [a,b], then we define
Theorem. If f(x) is integrable on a close interval containing the three nuber a, b, and c, then

40. Fundamental Integration
Example: Find the antiderivative of y(x) = 4x3 + 2x + 5
in the interval [2,5] of x.
Sol

41. Fundamental Integration
Exercises: Find the value of these definite integral. 1. 2. 3. 4. 5.

42. Additional applications of Integration
In Physics
Center of mass
The triangle plate has mass M and constant density ρ that is shown in the figure. Find its center of mass that can find from these equations H L

43. Additional applications of Integration
In Physics
Work done by a constant force
Work done by variable force ? F S

44. References Anton H., Bivens I., and Davis S. Calculus. 7th Ed.
John Willey&Son, Inc