LESSON 1 Mathematics for Physics Mr. Komsilp Kotmool (Aj Tae) Department of Physics, MWIT Email : [email protected] [email protected] Web site : tae_mwit.

LESSON 1LESSON 1Mathematics for PhysicsMathematics for Physics

Mr. Komsilp Kotmool (Aj Mr. Komsilp Kotmool (Aj Tae)Tae)

Department of Physics, Department of Physics, MWITMWIT

Email : Email : [email protected]

Web site : Web site : www.mwit www.mwit

.ac.th/~tae_mwit.ac.th/~tae_mwit

Warm UpWarm 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?

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.

CONTENTSCONTENTS

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 Why do we use Mathematics in Physics (and other subjects)?Physics (and other subjects)?

ข้�อมู�ลเชิงวิทยาศาสตร์� ข้�อมู�ลเชิงวิทยาศาสตร์� ((ฟิ�สกส�ฟิ�สกส�)) Qualitative Physics – เชิงคุ�ณภาพ เป็�นร์�บแบบเชิงคุ�ณภาพ เป็�นร์�บแบบ

การ์ศ"กษาท$%บ&งบอกถึ"งคุวิามูร์� �ส"ก มู$หร์)อไมู&มู$ การ์ศ"กษาท$%บ&งบอกถึ"งคุวิามูร์� �ส"ก มู$หร์)อไมู&มู$ เชิ&น บางส&วินข้องวิชิาเชิ&น บางส&วินข้องวิชิา Quantum Physics

Quantitative Physics – เชิงป็ร์มูาณ เป็�นร์�ป็แบบเชิงป็ร์มูาณ เป็�นร์�ป็แบบการ์บ&งชิ$+เชิงข้�อมู�ท$%แมู&นตร์ง มูากการ์บ&งชิ$+เชิงข้�อมู�ท$%แมู&นตร์ง มูาก--น�อยแคุ&ไหน น�อยแคุ&ไหน สามูาร์ถึน-าไป็ป็ร์ะย�กต�ใชิ�ในกจกร์ร์มูข้องมูน�ษย�สามูาร์ถึน-าไป็ป็ร์ะย�กต�ใชิ�ในกจกร์ร์มูข้องมูน�ษย�ได้�ย&างมู$ป็ร์ะสทธิภาพ ได้�ย&างมู$ป็ร์ะสทธิภาพ ((ส&วินใหญ่&ข้องฟิ�สกส�ส&วินใหญ่&ข้องฟิ�สกส�))

What is Mathematics in this What is Mathematics in this course?course?

REVIEW: basic problemDisplacement S = ? = v x t NOT AT ALLNOT AT ALL

condition: v must be constant !!But in reality, v is not constant !!!

General definition

Betterdt

dsv or vdts

dt

sdv

or dtvs

CALCULUSCALCULUS

FunctionsFunctions

What is FUNCTION?REVIEW : some basic equations

y = Ax+B lineary = Ax2+Bx+C parabolay = sin(x) trigonometricy = 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]

21 x

FunctionsFunctions

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 Note: We usually see f(x)f(x) and and g(x)g(x) in many text books. in many text books.

FunctionsFunctions

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 y is a function of x.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”)

ExerciseExercise

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

21 x

Limits and Continuity of functionLimits and Continuity of function

What is limits? Why do we use limits? When do we use limits?

??????????


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!!!!!!

2r

MmGF

0F

at r → ∞.


Electric field at arbitrary from solid sphere

E = ? At E = ? At r = ar = a


For mathematical equation

f(x=0) = ?11

)(

x

xxf

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

2)(lim0

xfx

“the limit of f(x) as x approaches 0 is 2”


How about at x → 0?xxxf /)(

xxxf /)(

-1 ; x<0

1 ; x>0 1)(lim0

xfx

Limit from the leftLimit from the left

1)(lim0

xfx

Limit from the rightLimit from the right

)(lim)(lim00

xfxfxx

)(lim0

xfx

Not exist !!For discontinuity


THEOREMS. Let a and k be real numbers, and suppose that

1)(lim Lxfax

and 2)(lim Lxgax

kkax

lim axax

lim

21)(lim)(lim)]()([lim LLxgxfxgxfaxaxax

and

21)(lim).(lim)]().([lim LLxgxfxgxfaxaxax

0,)(lim

)(lim]

)(

)([lim 2

2

1

L

L

L

xg

xf

xg

xf

ax

ax

ax

even. is if 0 ,)(lim)(lim 11 nLprovideLxfxf nnax

n

ax

1

2

3

4

5


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).

)(lim xfcx

)()(lim cfxfcx

History of CalculusHistory of Calculus

Sir Isaac Newton (1642-1727) Gottfried Wilhelm Leibniz (1646-1716)

Fundamental DerivativeFundamental DerivativeIn reality, the world phenomena involve changing quantities:In reality, the world phenomena involve changing quantities:

the speed of objects (rocket, vehicles, balls, etc.)the speed of objects (rocket, vehicles, balls, etc.)the number of bacteria in a culturethe number of bacteria in a culturethe shock intensity of an earthquakethe shock intensity of an earthquakethe voltage of an electrical signal the voltage of an electrical signal

It is very easy for the ideal situations {many exercises in your text books}

For example: constant of velocityS = v x t

rate of change of S is constant v = ΔS/Δt

Consider: can we have this situation in the real world?

Fundamental DerivativeFundamental Derivative

There are many conditions in nature affecting to complicate phenomena and equations!!!


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 tHow do we define velocity from t11 to t to t22 ? ?

Average velocity ?Average velocity ?

12

12 )()()(

tt

tStS

t

tSvav


Slopes and rate of changeS(t)

tt1

t2

S(t1)

S(t2)

slopett

tStSvav

12

12 )()(

vvavav can not be exactly can not be exactly

represented this motion!represented this motion!

Take t2 close to t1

S(t)

tt1 t2

S(t1)S(t2)

more exactly !!!

Clo

se to

tang

ent l

ine

Slope at t = tSlope at t = t11 is tangent line is tangent line


0 , )()()(

12

12

t

tt

tStS

t

tSvav

Take t2 close to t1 that t2-t1 = Δt → 0

ttt

tSttS

t

tSvv

ttav

100

int let ; )()(

lim)(

lim

vint → v(t) instantaneous velocity

Mathematical notationMathematical notation

dt

tdS

t

tSttSv

t

)(

)()(lim

0int

(read “(read “dee S by dee tdee S by dee t”)”)


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 DerivativeFundamental Derivative FORMULARS: If f(x) and g(x) are the function of x and c is any real

number

0)( cdx

d

1 nn nxxdx

d

xxdx

dcossin

xxdx

dsincos

1.

2.

3.

4.

dx

xdfcxcf

dx

d )()(

)()()()( xgdx

dxf

dx

dxgxf

dx

d

5.

6.


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:dx

xdu

du

udvxv

dx

d )(.

)()(


FORMULARS : If u is a function of x {u(x)}, and c is any real number

udx

dnuu

dx

d nn 1

udx

duu

dx

dcossin

udx

duu

dx

dsincos

1.

2.

3.

4.

5.

)()()()()().( xfdx

dxgxg

dx

dxfxgxf

dx

d

Frequency use in Physics

6.

7.

2)(

)()()()()(/)(

xg

xgdxd

xfxfdxd

xgxgxf

dx

d

udx

dee

dx

d uu

udx

d

uu

dx

d 1ln

g(x) ≠ 0

Additional applications of Additional applications of DerivativeDerivative

Maximum and Minimum problemsEx : An open box is composed of a 16-inch by 30-inch piece of 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 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 and bending up the sides. What size should the squares to be obtained a box with the largest volume?a box with the largest volume?

16 in

30 in

xx

xx

xx

xx

Fundamental IntegrationFundamental Integration

IntegrationIntegration การ์หาป็ร์พ4นธิ�การ์หาป็ร์พ4นธิ�What is the integration (calculus)?What is the integration (calculus)?

Why do we use the integration (calculus)?Why do we use the integration (calculus)?

When do we use the integration (calculus)?When do we use the integration (calculus)?


Recall:Recall: If we know S(t) of a object we can find v(t) of its and say say v(t)v(t) is the derivative of is the derivative of S(t)S(t) !!!!

dt

tdS

t

tSttStv

t

)(

)()(lim)(

0

Reverse problem :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)S(t) is the antiderivative of is the antiderivative of v(t)v(t) !!!!

What is the antiderivative?What is the antiderivative?


Consider graphs of constant and linear velocity with time (v(t)-t)

v(t)

t

v0

v(t)

t

v0

v

tt

S = vS = v00tt S = (1/2)(vS = (1/2)(v00+v)t+v)t

S(t) was represented by area under function of S(t) was represented by area under function of v(t)v(t) !!! !!!


How about the complicate function of v(t)?

v(t)

t

v

t

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?


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.

MMethod of ethod of EExhaustionxhaustion

disadvantage: do not proper with asymmetric shape

Fundamental IntegrationFundamental Integrationv(t)

t

v

tΔΔtt

t*

v(tv(t**))

ttvS ).( *

n

ii

n

ii ttvSS

1

*

1

).(

For smoother area nt 0

Therefore, we get

dttv

ttvttvtSn

ii

n

n

ii

t

)(

).(lim).(lim)(1

*

1

*

0

Read “S(t) equal integral of v(t) dee t”


Mathematical MMathematical Methodethod

For Polynomial Function

)()(

xfdx

xdF dxxfxF )()(

Note : f(x) is the derivative of F(x), but in the other hand, F(x) is a antiderivative of f(x).

F(x) = x2 + 5 f(x) = 2x

F(x) = x2 + 10 f(x) = 2x

F(x) = x2 f(x) = 2xConstants Constants depend on depend on conditionsconditions !!!!


Recall: 1 nn nxxdx

d

Set n=n-1 1 nn nxxdx

d nn xnxdx

d)1(1

Therefore, we get 1)1( nn xdxxn

1)1()1( nnn xdxxndxxn1

1

n

xdxx

nn

For the flexible form Cn

xdxx

nn

1

1

When C is the arbitrary constant.


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


cn

xdxx

nn

1

1

cxdxx ln1

cedxe xx

cxdxx tansec2

cxdxx sincos

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.

xgdxxgdx

d

dxxgcdxxcg

dxxgdxxfdxxgxf

cxdxx cotcsc2

cxdxx cossin

All of these are call the indefinite integral All of these are call the indefinite integral

Fundamental IntegrationFundamental IntegrationThe definite integral

dxxvxS )()( Recall:

We get the general solution of S(t) from the indefinite integralthe indefinite integral.

Problem: How do we get displacement (S) in the Problem: How do we get displacement (S) in the interval [a,b] of time? interval [a,b] of time?

v(t)

ta b

a is the lower limitb is the upper limit

)()()()( aSbStSdttvb

a

b

a

this is the definite integral !this is the definite integral !

Fundamental IntegrationFundamental IntegrationDefinition: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.Theorem. If f(x) is integrable on a close interval containing the three nuber a, b, and c, then

0)( a

a

dxxf

b

a

a

b

dxxfdxxf )()(

b

c

c

a

b

a

dxxfdxxfdxxf )()()(

Fundamental IntegrationFundamental IntegrationExample: Find the antiderivative of y(x) = 4x3 + 2x + 5 in the interval [2,5] of x.

645

)25(5)25()25(

5

10

5

11

2

13

4

524

)524()(

2244

5

2

5

2

25

2

4

5

2

105

2

115

2

13

5

2

5

2

5

2

3

5

2

35

2

xxx

xxx

dxdxxdxx

dxxxdxxySolSol


Exercises: Find the value of these definite integral.

1.

2.

3.

4.

5.

0

sin d

1

0

5)53( dxx

24

0

1 sin

dxxx

0

2 5sec ydy

3/

0

tansec

d

Additional applications of Additional applications of IntegrationIntegrationIn Physics

Center of mass

HH

LL

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

xdmx Mcm1

ydmy Mcm1

Additional applications of Additional applications of IntegrationIntegrationIn PhysicsIn Physics

Work done by variable force ?

FSW Work done by a constant force

F

S

FdSW

ReferencesReferences

Anton H., Bivens I., and Davis S. Calculus. 7th Ed. John Willey&Son, Inc. 2002.

LESSON 1 Mathematics for Physics Mr. Komsilp Kotmool (Aj Tae) Department of Physics, MWIT Email : [email protected] [email protected] Web site : tae_mwit.

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