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LESSON 1
Mathematics for Physics
Mr. Komsilp Kotmool (Aj Tae)
Department of Physics, MWIT
Email : amolozo@hotmail.com
Web site : www.mwit .ac.th/~tae_mwit
Warm Up

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?
the farmer needs 3 hens to
produce 12 eggs in 6 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?
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
Why do we use Mathematics in
Physics (and other subjects)?
 ข้อมูลเชิงวิทยาศาสตร์ (ฟิ สก
ิ ส์)

Qualitative Physics – เชิงคุณภาพ เป็ นรูบแบบการศึกษา
ทีบ่ ง่ บอกถึงความรูส้ กึ มีหรือไม่มี เช่น บางส่วนของวิชา
Quantum Physics

Quantitative Physics – เชิงปริมาณ
เป็ นรูปแบบการบ่งชี้
เชิงข้อมูทแ่ี ม่นตรง มาก-น้อยแค่ไหน สามารถนาไปประยุกต์ใช้
ในกิจกรรมของมนุษย์ได้ยา่ งมีประสิทธิภาพ (ส่วนใหญ่ของ
ฟิ สกิ ส์)
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
ds
v
dt
or

 ds
v
dt
or
s   vdt


s   v dt
CALCULUS
Functions
What is FUNCTION?
REVIEW : some basic equations
y = Ax+B
y = Ax2+Bx+C
y = sin(x)
y =  1 x2


linear
parabola
trigonometric
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]
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.
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”)
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 + 1  x 2 = 0
4). ycsc(x) = 1
5). y2 +2y –x =0
Limits and Continuity of function



What is limits?
Why do we use limits?
When do we use limits?
?????
Limits and Continuity of function

Newton’s Law of Universal Gravitation

Mm
F G 2
r
We can not calculate force at r → ∞.
How do we solve this problem??
But we can find the close value!!!!!!

F 0
at r → ∞.
Limits and Continuity of function

Electric field at arbitrary from solid sphere
E = ? At r = a
Limits and Continuity of function

For mathematical equation
f ( x) 
x
x  1 1
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
lim f ( x )  2
x 0
“the limit of f(x) as x approaches 0 is 2”
Limits and Continuity of function

How about f ( x)  x / x
at x → 0?
-1 ; x<0
f ( x)  x / x 
1 ; x>0
lim f ( x)  1
x 0
lim f ( x)  1
Limit from the left
Limit from the right
x 0
lim f ( x)  lim f ( x)
x 0
x 0
lim f ( x ) Not exist !!
x 0
For discontinuity
Limits and Continuity of function
THEOREMS. Let a and k be real numbers, and suppose that

lim f ( x)  L1
and
xa
lim g ( x)  L2
xa
1
lim k  k
2
lim [ f ( x)  g ( x)]  lim f ( x)  lim g ( x)  L1  L2
3
lim [ f ( x).g ( x)]  lim f ( x). lim g ( x)  L1 L2
4
f ( x) L
f ( x) lim
x a
lim [
]
 1 , L2  0
x a g ( x)
lim g ( x) L2
xa
lim x  a
and
xa
x a
x a
x a
x a
x a
xa
x a
5
lim
x a
n
f ( x)  n lim f ( x)  n L1 , provide L1  0 if n is even.
x a
Limits and Continuity of function

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). lim f ( x ) exists
x c
3). lim f ( x)  f (c)
x c
History of Calculus
Sir Isaac Newton (1642-1727)
Gottfried Wilhelm Leibniz (1646-1716)
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=vxt
rate of change of S is constant v = ΔS/Δt
Consider: can we have this situation in the real world?
Fundamental Derivative
There are many conditions in nature affecting to complicate phenomena
and equations!!!
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 ?
S (t ) S (t 2 )  S (t1 )
vav 

t
t 2  t1
Fundamental Derivative
Slopes and rate of change
S(t)
Take t2 close to t1
S(t)
vav 
S (t 2 )  S (t1 )
 slope
t 2  t1
S(t2)
S(t2)
S(t1)
S(t1)
t
t1
t2
vav can not be exactly
represented this motion!
t
t1 t2
more exactly !!!
Slope at t = t1 is tangent line
Fundamental Derivative
Take t2 close to t1 that t2-t1 = Δt → 0
vav 
S (t ) S (t 2 )  S (t1 )

, t  0
t
t 2  t1
S (t )
S (t  t )  S (t )
 lim
; let t1  t
t 0 t
t 0
t
vav  vint  lim
vint → v(t) instantaneous velocity
Mathematical notation
S (t  t )  S (t ) dS (t )

t 0
t
dt
vint  lim
(read “dee S by dee t”)
Fundamental Derivative
Exercises: Find the derivative of y(x) and its value at x=2
1. y(x) = x
2. y(x) = x2
3. y(x) = sin(x)
Fundamental Derivative

FORMULARS: If f(x) and g(x) are the function of x and c is any real
number
d
d
5.
(
c
)

0
sin x   cos x
1.
dx
dx
 
2.
d n
x  nx n 1
dx
3.
d
cf ( x)   c df ( x)
dx
dx
4.
d
 f ( x)  g ( x)   d f ( x)  d g ( x)
dx
dx
dx
6.
d
cos x    sin x
dx
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:
d
dv(u ) du ( x)
v( x) 
.
dx
du
dx
Fundamental Derivative

FORMULARS : If u is a function of x {u(x)}, and c is any real
number
1.
 
d n
n 1 d
u  nu
u
dx
dx
Frequency use in Physics
d
sin u   cos u d u
dx
dx
6.
d
cos u    sin u d u
3.
dx
dx
7.
2.
4.
5.
d u
e  eu
dx
d
1
ln u 
dx
u
d
u
dx
d
u
dx
d
 f ( x).g ( x)  f ( x) d g ( x)  g ( x) d f ( x)
dx
dx
dx
d
 f ( x) / g ( x)  
dx
g ( x)
d
d
f ( x)  f ( x) g ( x)
dx
dx
2
g ( x)
g(x) ≠ 0
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?
x
x
16 in
x
x
30 in
Fundamental Integration
Integration
การหาปริพนั ธ์
What is the integration (calculus)?
Why do we use the integration (calculus)?
When do we use the integration (calculus)?
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) !!
S (t  t )  S (t ) dS (t )
v(t )  lim

t 0
t
dt
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?
Fundamental Integration
Consider graphs of constant and linear velocity
with time (v(t)-t)
v(t)
v(t)
v
v0
v0
S = v 0t
t
t
S = (1/2)(v0+v)t
t
t
S(t) was represented by area under function of v(t) !!!
Fundamental Integration
How about the complicate function of v(t)?
v(t)
What about S(t) of this curve?
v
S(t) also was represented
by area under of this curve!
t
t
Next problem….
How do we find this area?
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
Fundamental Integration
v(t)
S  v(t ).t
*
n
n
i 1
i 1
S   Si   v(ti ).t
v
*
For smoother area
t  0  n  
v(t*)
Therefore, we get
t
Δt
t*
t
S (t )  lim
t 0
n
 v(ti ).t  lim
i 1
*
n 
n
 v(ti ).t
*
i 1
  v(t )dt
Read “S(t) equal integral of v(t) dee t”
Fundamental Integration
Mathematical Method
dF ( x)
F ( x)   f ( x)dx
 f ( x)
dx
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
F(x) = x2 + 5
f(x) = 2x
F(x) = x2 + 10
f(x) = 2x
Constants
depend on
conditions !!
Fundamental Integration
Recall:
Set n=n-1
 
d n
x  nx n 1
dx
d n
x  nx n 1
dx
 
Therefore, we get
n
n 1
(
n

1
)
x
dx

x

n
n
n 1
(
n

1
)
x
dx

(
n

1
)
x
dx

x


For the flexible form
 
d n 1
x
 (n  1) x n
dx
n 1
x
n
x
 dx  n  1
n 1
x
n
x
 dx  n  1  C
When C is the arbitrary constant.
Fundamental Integration
Exercises: Find the antiderivative of the following y(x).
1. y(x) = 3x3 + 4x2 + 5
2. y(x) = 0
3. y(x) = sin(x)
4. y(x) = xcos(x2)
5. y(x) = tan(x)
Integration by
Substitution
Fundamental Integration

FORMULARS: If f(x) and g(x) are the function of x and c is
any real number
x n 1
c
n 1
1.
 x dx
2.
1
x
 dx  ln x  c
n

6.
2
sec
 x dx  tan x  c
7.
2
csc
 x dx   cot x  c
 g xdx
3.
 sin x dx   cos x  c 8.
d
dx
4.
 cos x dx
 cg x dx
5.
x
x
e
dx

e
c

 sin x  c
9.

g x 
 c  g x dx
10.   f x   g x dx   f x dx   g x dx
All of these are call the indefinite integral
Fundamental Integration
The definite integral
Recall:
S ( x)   v( x)dx
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)
this is the definite integral !
b
b
a
a
 v(t )dt  S (t )
a
b
t
 S (b)  S (a)
a is the lower limit
b is the upper limit
Fundamental Integration
Definition:
(a) If a is in the domain of f(x), we define
a
 f ( x)dx  0
a
(b) If f(x) is integrable on [a,b], then we define
a
b
b
a
 f ( x)dx   f ( x)dx
Theorem. If f(x) is integrable on a close interval containing
the three nuber a, b, and c, then
b
c
b
a
a
c
 f ( x)dx   f ( x)dx   f ( x)dx
Fundamental Integration
Example: Find the antiderivative of y(x) = 4x3 + 2x + 5
in the interval [2,5] of x.
Sol
5
5
2
2
3
y
(
x
)
dx

(
4
x

  2 x  5 )dx
5
5
5
2
2
2
  4 x 3 dx   2 x dx   5 dx
31 5
11 5
0 1 5
4x
2x
5x



3 1 2 11 2 0 1
x
4 5
2
x
2 5
2
2
 5x 2
5
 (54  2 4 )  (52  2 2 )  5(5  2)
 645
Fundamental Integration
Exercises: Find the value of these definite integral.

1.
 sin d
0
1
5
(
3
x

5
)
dx
2. 
0
4 2
3.

1
x
sin x dx
0

2
sec
 5 ydy
4.
0
 /3
5.
 sec  tan d
0
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
xcm 
1
M
ycm 
1
M
 xdm
 ydm
H
L
Additional applications of
Integration
In Physics
Work done by a constant force
W  FS
Work done by variable force ?
F
W   FdS
S
References
Anton H., Bivens I., and Davis S. Calculus. 7th Ed.
John Willey&Son, Inc. 2002.