math physics

5/20/2018 Math Physics

1/75

I(Basic Mathematics for Physics I)

Suppiya Siranan

[email protected]

.http://physics2.sut.ac.th/~suppiya/note/Mathematics01.pdf

Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 1/75
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2/75

Physical laws should have mathematical beauty.

PAUL ADRIEN MAURICE DIRAC( )The Nobel Laureate in Physics 1933

: 8 ..1902 Bristol,England: 20 ..1984Tallahassee, Florida,USAhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.html

http://en.wikipedia.org/wiki/Paul_Dirachttp://nobelprize.org/nobel_prizes/physics/laureates/1933/

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3/75

(empty set) (element)

(natural number)(positive integer)

N Z+

1, 2, 3, 4, 5, . . .

(negative integer) Z

1, 2, 3, 4, 5, . . .

(integer) Z

. . . , 3, 2, 1, 0, 1, 2, 3, . . .

= Z+ Z

0

(0 Z0 / Z+ 0 / Z)

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4/75

(rational number)

Q

x x= a

b a

Z

b

Z

02 (irrational number)

Q

x x = a

b

aZ b

Z

0 2 , e,2 Q

Q =

(real number) R Q Q

(complex number)

C z z =x+ iy x, y R i 1 Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 4/75


5/75

I

(Factorial & Binomial Theorem)



6/75

Factorial

(factorial)n n Z+

0() n! (recurrencerelation)

n! 1 n= 0n (n 1)! n Z+ (1)

0! 1 n! 1 2 3 n n Z+ (2) (calculus)

n! = 0

tn et dt n Z+ 0 (3) (3) nn R n Z



7/75

Binomial Theorem

(Binomial Series)(a+b)1 =a+b

(a+b)2 =a2 + 2ab+b2

(a+b)3 =a3 + 3a2b+ 3ab2 +b3

(a+b)4 =a4 + 4a3b+ 6a2b2 + 4ab3 +b4

.

..

(Pascals Triangle)1 1

1 2 11 3 3 1

1 4 6 4 1

... (Binomial Theorem) n Z+

(a+b)n =n

k=0

nkankbk =an + n

1an1b+ n

2an2b2 +. . . (4)

n

k n!(n k)! k! (binomial coefficient)Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 7/75


8/75

n

k

=

n(n 1)(n 2) . . . (n k+ 1)k!

(5)

(5) n n (n R) (Binomial Theorem) n

R

(1 +x)n =k=0

n

k

xk

= 1 +nx+ n(n 1)2

x2 + n(n 1)(n 2)6

x3 +. . . (6)

1< x 1 x 1, 1 a > b (a+b)n (a + b)n =an 1 +n

b

a+n(n 1)

2 b

a2

+n(n 1)(n 2)

6 b

a3

+. . .Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 8/75


9/75

BLAISE PASCAL

( )

: 19 ..1623Clermont (Clermont-Ferrand),

Auvergne,France: 19 ..1662Paris,France

http://www-history.mcs.st-andrews.ac.uk/Biographies/Pascal.html

http://en.wikipedia.org/wiki/Blaise_Pascal

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10/75

: 11

11 = (9 + 2)1/2 = 91/21 +291/2

11 = 31 +

1

2 2

9+1

2 1

21

21

2

92

+1

6

1

2

1

2 1

1

2 2

2

9

3+. . .

= 3

1 +

1

2

2

9

1

8

2

9

2+

1

16

2

9

3 5

128

2

9

4+. . .

= 3 +1

3 1

54+

1

486 5

17 496+. . .

11 = 3.316 . . .



11/75

II

(Summation Formulae)



12/75

Summation Formulaen

k=1

k= 1 + 2 + 3 + 4 +. . . +n

n

k=1

k=n+ (n 1) + (n 2) + (n 3) +. . . + 1

2

n

k=1k= (n+ 1) + (n+ 1) + (n+ 1) + (n+ 1) +. . . + (n+ 1)

n=n(n+ 1)

n

k=1 k= n(n+ 1)

2

(7)



13/75

(k+ 1)3 k3 =k3 + 3k2 + 3k+ 1 k3 = 3k2 + 3k+ 1

n

k=1

(k+ 1)3 k3=n

k=1

3k2 + 3k+ 1 n

k=1 (k+ 1)3

k3= (n+ 1)

3

n3+ n

3

(n

1)3+. . .

+

43 33+ 33 23+ 23 13

2 n

k=1 (k+ 1)3 k3= (n+ 1)3 1

telescoping sumSuppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 13/75


14/75

(n+ 1)

3

1 =n

k=1

3k2 + 3k+ 1= 3

n

k=1 k2 + 3

n

k=1 k+n

k=1 1= 3

n

k=1k2 + 3

n(n+ 1)

2 +n

3

n

k=1 k2 = (n+ 1)3 1 3n(n+ 1)

2 n

n

k=1 k2 = n(n+ 1)(2n+ 1)

6

(8)



15/75

nk=1

(k+ 1)4 k4

(7) (8) nk=1

k3 =

n(n+ 1)

2

2(9)

nk=1

(k+ 1)5 k5 (7), (8)

(9) n

k=1

k4 = n(n+ 1)(2n+ 1)(3n2 + 3n 1)

30 (10)

nk=1

(k+ 1)6 k6 (7)(10)

nk=1

k5 = n

2

(n+ 1)

2

(2n

2

+ 2n 1)12 (11)Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 15/75


16/75

III

(Exponential & LogarithmFunctions)


E ti l F ti


17/75

Exponential Function

(exponential function) ax a R+ x R a (base) x (exponent)

(natural logarithmic base)e

e limn1 + 1nn

limm0 1 +m1/m (12)

1 + 1nn = 1 + n

1! 1

n+ n(n 1)

2! 1

n2 + n(n 1)(n 2)

3! 1

n3 +. . .

= 1 + 1

1!

+ 1

2! 1 1

n+ 1

3! 1 1

n1 2

n+. . .Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 17/75


18/75

e =k=0

1k!

= 1 + 11!

+ 12!

+ 13!

+ 14!

+. . .= 2.718281828459045 . . . (13)

ex = lim

1 +1x = lim

1 +1

x

n=x

1

=

x

n

n

exp x ex = limn1 + xnn (14)

ex =

k=0

xk

k! = 1 +x+x2

2! +x3

3! +x4

4! +. . . (15)Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 18/75

L ith F ti


19/75

Logarithm Function

(logarithmic function) a R+

x R+

logax

y= loga

x

ay =x

(16)y = loga

ay

a(logax) =x (17)

(natural logarithmic function) a e ln x= logex

y = ln x ey =x (18)

y= ln ey= ln(exp y) exp(ln x) = e(lnx)

=x (19)Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 19/75


20/75

a0 = 1

ax = 1

ax

axay =ax+y

ax

ay =axy

a(x logay) =yx

logax+ logay = loga(xy)

logax logay = loga

x

y

logax

logay = logyx

loga

yx

=x logay

x,y,a R+



21/75

IV

(Trigonometry)


Trigonometry


22/75

Trigonometry O(0, 0)

() x

(radian)

s

r

r = 1 s = 90 =

2 rad,

180

= rad 360 = 2 rad

x

y

r s

O(0, 0)

P(x, y)



23/75

(trigonometric functions)

1. (cosine & sine functions) (cos , sin ) (x, y) cos(+ 2n) = cos sin(+ 2n) = sin n Zcos() = cos(2n ) = cos sin() = sin(2n ) = sin n Zcos2 + sin2 = 1 cos2 (cos )2 sin2 (sin )2

2. (secant & cosecant functions)

sec

1

cos csc

1

sin

3. (tangent & cotangent func-tions)

tan sin

cos cot cos

sin = 1

tan



24/75
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25/75

V

(Differential Calculus)


Derivative


26/75

Derivative

f(x) x

f(x) ddx

f(x) limh0

f(x+h) f(x)h

(20)

f(x) f(x) x (derivativeof function f(x)with respect to x)

f(x) x x=x0

f(x0) ddx

f(x)

x=x0

limh0

f(x0+h) f(x0)h

(21a)

f(x0) ddx

f(x)

x=x0 lim

xx0

f(x) f(x0)x

x0

(21b)



27/75

1. cf(x) c

d

dx

cf(x)

= lim

h0

cf(x+h) cf(x)h

=c limh0

f(x+h) f(x)h

d

dx

cf(x)

=c

d

dxf(x) (22)

2. () f(x)

g(x)

d

dx

f(x) g(x)= lim

h0

f(x+h) g(x+h) f(x) g(x)

h

= limh0 f(x+h) f(x)h g(x+h) g(x)h d

dx f(x) g(x)= d

dxf(x)

d

dxg(x) (23)


3. f(x) g(x)
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28/75

( ) ( )

d

dx f(x) g(x)= limh0f(x+h) g(x+h) f(x) g(x)

h

= limh0

f(x+h) g(x+h) f(x) g(x+h) +f(x) g(x+h) f(x) g(x)h

= limh0

f(x+h) f(x)h

g(x+h) +f(x) g(x+h) g(x)h

= limh0

f(x+h) f(x)h g(x) +f(x) limh0

g(x+h) g(x)h

d

dx f(x) g(x)

=

d

dxf(x)

g(x) +f(x)

d

dxg(x)

(24)


4. f(x)/g(x) g(x) = 0
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29/75

d

dx f(x)

g(x) = limh01

h f(x+h)

g(x+h)f(x)

g(x) = lim

h0

f(x+h) g(x) f(x) g(x+h)h g(x+h) g(x)

= limh0

f(x+h) g(x) f(x) g(x) +f(x) g(x) f(x) g(x+h)h g(x+h) g(x)

= limh0 f(x+h) f(x)

h g(x) f(x) g(x+h) g(x)

h g(x+h) g(x)

ddxf(x)

g(x)

= ddx f(x) g(x) f(x) ddx g(x)

g(x)2 g(x) = 0 (25)


5. fg(x) f(z) z g(x)
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30/75

z+k =g(x+h) h 0k 0

ddx

f

g(x)

= limh0

fg(x+h) fg(x)h

= limh0(k0)

f(z+k) f(z)

h

k=g(x+h)

g(x)

= limh0(k0)

f(z+k) f(z)

k k

h

=

limk0

f(z+k) f(z)k

limh0

g(x+h) g(x)h

ddx

fg(x)= ddz

f(z) ddx

g(x) z g(x) (26) (26) (chain rule)


I
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31/75

d

dx c= 0

d

dxxn =nxn1

ddx

ex = ex

d

dx|ln x| = 1

x x = 0

ddx

ax =ax ln a

d

dx|logax| =

1

x ln a x = 0

d

dxsin x= cos x

d

dxcos x= sin x

ddx

tan x= sec2 x

d

dxsec x= sec x tan x

ddx

csc x= csc x cot xd

dxcot x= csc2 x


II


32/75

ddx

sin1 x= 11 x2 (|x| 1)

ddx

cot1 x= 11 +x2


Higher Order Derivatives


33/75

Higher Order Derivatives

(higher order derivatives) (second order derivative)

f(x) d2

dx2f(x) d

dx

ddx

f(x)

limh0

f(x+h) f(x)h

(27)

(third order derivative)

f(x) d3

dx3f(x) d

dx d2

dx2f(x)

lim

h0

f(x+h) f(x)h

(28)

f(4)(x), f(5)(x), f(6)(x), . . . f(x), f(x), f(x), . . . !!


n


34/75

n (LeibnizsFormula fornth Derivative of a Product)

Leibnizs Formula fornth Derivative of a Product:dn

dxn f(x)g(x)

=

n

k=0

n

k

dnk

dxnkf(x)

dk

dxkg(x)

= dn

dxnf(x) +n

dn1

dxn1f(x)

d

dxg(x)

+ n(n 1)2! dn2dxn2 f(x) d2dx2 g(x)+. . .+

dn

dxng(x) (29)


GOTTFRIED WILHELM VON LEIBNIZ ( LEIBNITZ)
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35/75

GOTTFRIED WILHELM VON LEIBNIZ (LEIBNITZ)( )

: 1 ..1646Leipzig, Saxony,Germany

: 14 ..1716Hannover,Germany

http://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.html

http://en.wikipedia.org/wiki/Gottfried_Leibniz


Partial Derivatives
http://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www.sut.ac.th/http://www.sut.ac.th/http://physics3.sut.ac.th/http://physics2.sut.ac.th/~suppiyahttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.htmlhttp://physics3.sut.ac.th/http://physics3.sut.ac.th/


36/75

Partial Derivatives 2 2 f(x, y) x y

fx(x, y) x

f(x, y) limh0

f(x+h, y) f(x, y)h

(30)

fx(x, y) f(x, y) x

(partial derivative of function f(x, y)with respect to x) f(x, y) y x

fy(x, y) y

f(x, y) limk0

f(x, y+k) f(x, y)k (31)

fy(x, y) f(x, y) y

(partial derivative of function f(x, y)with respect to y)Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 36/75

(second order partial derivatives) 4


37/75

x

fxx(x, y) 2

x2f(x, y)

x

x

f(x, y)

(32a)

y

fyy(x, y)

2

y2f(x, y)

y

yf(x, y)

(32b)

xyfxy(x, y)

2

yxf(x, y)

y

xf(x, y)

(32c)

y xfyx(x, y)

2

xyf(x, y)

x

yf(x, y)

(32d)



38/75

VI

(Integral Calculus)


Indefinite Integral (Antiderivative)


39/75

g ( )

(indefinite integral)(antideriva-tive)f(x) =

F(x) dx F(x) d

dxf(x) (33)

f(x)(antiderivative) F(x)

d

dx F(x) dx= F(x) (34a)

ddx f(x) dx=f(x) +C (34b) C (arbitrary constant)



41/75

(definite integral) (Rie-mann Sum)

b

a

F(x) dx

limmaxxi0(n)

n

i=1 F(xi ) xi (35)a, b n xi xi xi1 x0 a xn b

ni=1

xi = (b a)

xi i xi1 xi xi xi xi1, xi

x xi = (b a)/n xii xi =xi1 xi =xi

xi = (xi1+xi)/2Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 41/75

xi =a+ix=a+ i

(b a) (36)


42/75

n( ) ( )

ba

F(x) dx= (b a) limn

1n

ni=1

F(xi ) (37)

xi

xi =a+ib a

n (38a)

xi =a+ (i 1)b a

n (38b)

xi =a+ i 12b an (38c)


GEORG FRIEDRICH BERNHARD RIEMANN
http://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www.sut.ac.th/http://www.sut.ac.th/http://physics3.sut.ac.th/http://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.htmlhttp://physics3.sut.ac.th/


43/75

( )

: 17 ..1826Breselenz, Hanover,Germany

: 20

..1866

Selasca,Italy

http://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.html

http://en.wikipedia.org/wiki/Bernhard_Riemann


: 31

x2 dx
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44/75

1

xi xi =a+ib an = 1 +2in

3

1

x2 dx= 2 limn

1

n

n

i=1 1 +2i

n 2

= 2 limn

1

n

n

i=1 1 +4i

n

+4i2

n

2 = 2 lim

n

1

n

n

i=11 +

4

n

n

i=1i+

4

n2

n

i=1i2

= 2 limn

1

n

n+

4

n

n(n+ 1)

2 +

4

n2n(n+ 1)(2n+ 1)

6


3x2 dx = 2 lim

1 + 2

n(n+ 1)+

2 n(n+ 1)(2n+ 1)


45/75

1

x dx 2 limn

1 + 2

n2 +

3 n3

= 2 limn

1 + 2

1 +

1

n

+

2

3

1 +

1

n

2 +

1

n

= 2 limn1 + 2 +43

3

1

x2 dx=26

3

ddx

x3

3

=x2 x

3

3

x=3

x3

3

x=1

= 27 13

= 26

3

31

x2 dx= x3

3

x=3

x3

3

x=1

. . . Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 45/75

(Fundamental Theorem ofCalculus) 2


46/75

Calculus) 2

Fundamental Theorem of Calculus I:F(x)a, b f(x)F(x)

a, b

ba

F(x) dx=f(b) f(a) f(x)bx=a

(39)

b

a d

dx

f(x) dx= f(b) f(a) Fundamental Theorem of Calculus II: F(x) I F(x) I

a I f(x) = xa F(t) dt F(x) I:

d

dx f(x) =

d

dx x

a F(t) dt=F(x) (40)Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 46/75


47/75

(Leibnizs Rule forDifferentiating an Integral)

Leibnizs Rule for Differentiating an Integral:

F(x, t) b(x)a(x)

F(x, t) dt I a(x), b(x) I

ddx b(x)a(x) F(x, t) dt= b(x)

a(x) x F(x, t) dt

+F(x, b)db(x)

dx F(x, a)

da(x)

dx (41)



48/75

VII

(Taylor & Maclaurin Series)


Taylor & Maclaurin Series


49/75

(Taylor series)f(x)

x=a

k=0(x a)k

k! f(k)(a) =f(a) + (x a)f(a) +(x a)

2

2! f(a)

+(x a)3

3! f(a) +

(x a)44!

f(4)(a) +. . . (42)

a = 0

f(x)

x = 0

(Maclaurin series) f(x)

k=0xk

k!f

(k)

(0) =f(0) +x f

(0) +

x2

2!f

(0) +

x3

3!f

(0)

+x4

4!f(4)(0) +

x5

5!f(5)(0) +

x6

6!f(6)(0) +. . . (43)


BROOK TAYLOR

( )
http://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www.sut.ac.th/http://www.sut.ac.th/http://physics3.sut.ac.th/http://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://physics3.sut.ac.th/


50/75

( )

: 18 ..1685Edmonton, Middlesex,England

:

29 ..1731Somerset House,London,England

http://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.html

http://en.wikipedia.org/wiki/Brook_Taylor


COLIN MACLAURIN( )
http://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://en.wikipedia.org/wiki/Brook_Taylorhttp://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www.sut.ac.th/http://www.sut.ac.th/http://physics3.sut.ac.th/http://physics2.sut.ac.th/~suppiyahttp://en.wikipedia.org/wiki/Brook_Taylorhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://en.wikipedia.org/wiki/Brook_Taylorhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.htmlhttp://physics3.sut.ac.th/http://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://physics3.sut.ac.th/


51/75

( )

: ..1698

Kilmodan (

Tighnabruaich 12), Cowal, Argyllshire,Scotland

: 14 ..1746Edinburgh,Scotland

http://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.html

http://en.wikipedia.org/wiki/Colin_Maclaurin


f(x)
http://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://en.wikipedia.org/wiki/Colin_Maclaurinhttp://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www.sut.ac.th/http://www.sut.ac.th/http://physics3.sut.ac.th/http://physics2.sut.ac.th/~suppiyahttp://en.wikipedia.org/wiki/Colin_Maclaurinhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://en.wikipedia.org/wiki/Colin_Maclaurinhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Maclaurin.htmlhttp://physics3.sut.ac.th/http://physics3.sut.ac.th/


52/75

x=a f(x)

f(x) x= a f(x)

f(x) = ex x R

ex =

k=0xk

k!

= 1 +x+x2

2!

+x3

3!

+x4

4!

+x5

5!

+. . . (44)

f(x) = ln(1 +x) 1< x 1

ln(1 +x) =k=1

(1)k1k

xk =x x2

2 +

x3

3 x

4

4 +. . . (45)

ln 2 = 1 1

2+

1

31

4+. . . (converge) Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 52/75

f(x) = cos x x R x (radian)


53/75

(radian)

cos x=k=0

(1)k(2k)!

x2k = 1 x2

2! +

x4

4! x

6

6! +. . . (46)

f(x) = sin x x R x (radian)

sin x=

k=0

(

1)k

(2k+ 1)!x2k+1 =x

x3

3! +x5

5!x7

7! +. . . (47)

f(x) = 1/(1 +x)

1< x


54/75

tan1

x=

k=0

(

1)k

(2k+ 1)!x2k+1

=x x3

3 +

x5

5 x7

7 +. . . (49)

4 = 1 1

3+

1

5 1

7+. . . (converge)

(45)

ln(1 +x) =x

x2

2 +

x3

3 x4

4 +

x5

5 x6

6 +. . .

1< x 1 xx

ln(1

x) =

x

x2

2 x3

3 x4

4 x5

5 x6

6 . . .

1 x


55/75

ln z z R+

ln z= 2

x+x3

3 +

x5

5 +

x7

7 +. . .

x= z 1

z+ 1 (50)

: ln 2

z = 2

x =

2 12 + 1

= 1

3

ln 2 = 2

1

3+

1

3 1

33

+1

5 1

35

+1

7 1

37

+. . .

= 2

1

3+

1

81+

1

1215+

1

15 309+. . .

= 0.693 147 . . .

34 3



56/75

VIII

(Scalar & Vector)


Scalar & Vector


57/75

(scalar) (magnitude) (temperature), (mass), (time), (energy)

(vector) (magnitude) (direction) (displace-ment), (velocity), (acceleration), (force), (momentum)

A, A, A, A

, A(), A(), A, A

A

A

,A

ASuppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 57/75

1 A B
http://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www.sut.ac.th/http://www.sut.ac.th/http://physics3.sut.ac.th/http://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/


58/75

1. A B

A=

B

2. (zero vector) 0

0= 03. Am mA mAA

(a) m >0 A = 0 m

A

A

(b) m


59/75

A (

A =

0) AA=

1

AA=

1

AA (52)

A

A=A A

(53)5. (x,y,z) A

(components) x, y z

A=Ax +Ay +Azk (54)

,

k

x

,y

z


Scalar Product (Dot Product)


60/75

(scalar productdot product) A B A B

A B ABcos = B A (55)

A= A, B = B A B = = k k= 1 = k=k = 0

AB x, yz A=Ax + Ay + Azk B=Bx + By + Bzk

A B=

Ax +Ay +Azk

Bx +By +Bzk

A B=AxBx+AyBy+AzBz (56)


Vector Product (Cross Product)


61/75

(vector productcross product)A B A B

A B ABsin n= B A (57)

A= A, B = B, A B n A B A B

A B

n

A

BB A

n

A B


= =kk= 0 = =k, k= k= k = k=


62/75

AB x, yz A=Ax + Ay + Azk B=Bx + By + Bzk

A B= Ax +Ay +Azk Bx +By +Bzk

A B= AyBz AzBy + AzBx AxBz+

AxBy AyBxk (58a) (determinant)

A B=

k

Ax Ay Az

Bx By Bz

(58b)


Gradient, Divergence & Curl

( fi ld) A() A( )


63/75

(vector field) A(r) A(x,y,z)

A(r) =Ax(r) +Ay(r) +Az(r) k

(scalar field) (r) (x,y,z)

(del operator nabla)

x

+ y

+ z

k (59)

(gradient) =(r)

x +

y +

zk (60)


(divergence) A= A(r)


64/75

A Axx

+ Ayy

+ Azz (61)

(curl) A= A(r)

A

yAz

zAy

+

z

Ax x

Az

+ x Ay y Ax k (62a)

A=

k

x

y

z

Ax Ay Az

(62b)



65/75

IX

(Complex Number)


Complex Number

2 + 1 0 R


66/75

x2 + 1 = 0 xR

x2 + 1 = 0

(complex number) z

z=x+ iy i 1 (63)

x

y

(x, y

R

) i

(imaginaryunit) i2 = 1 C

C z z =x+ iy x, y R i 1 (64)

x (real part) z (z)Re(z) y (imaginary part) z (z) Im(z)


2 a+ ibc+ id


67/75

(a+ ib) + (c+ id) = (a+c) + i(b+d)(a+ ib) (c+ id) = (a c) + i(b d)(a+ ib)(c+ id) = (ac

bd) + i(bc+ad)

(a+ ib)

(c+ id)=

(a+ ib)(c id)(c+ id)(c id) =

ac+bdc2 +d2

+ ibc ad

c2 +d2

(complex conjugate) z =x+ iy

z

z x iy (65)

Re(z) =

z+z

2 Im(z) = z z

2i (66)


(modulus) z |z|(|z| 0)


68/75

|z| x2 +y2 = (x+ iy)(x iy) (67)|z| = zz = zz

z =x+ iy (x, y)(xy)


z =x+ iy (r, ) x r cos y r sin


69/75

x=r cos y=r sin (polar form of complex number)

z =rcos + i sin r (68) r=

x2 +y2 = tan1

yx

z =r

cos(+ 2n) + i sin(+ 2n)

n Z (69)


Euler Formula


70/75

ei = 1 + (i) +

(i)2

2! +

(i)3

3! +

(i)4

4! +

(i)5

5! +

(i)6

6! +

(i)7

7! +. . .

= 1 + i 2

2! i3

3! +

4

4! + i

5

5!6

6! i7

7! +. . .

=

1

2

2! +

4

4!

6

6! +. . .

cos +i

3

3! +

5

5!

7

7! +. . .

sin (Euler formula)

ei = cos + i sin (70) (exponential function) (trigonometric functions)



71/75

(Euler form of complex number)z =rei =rei(+2n) n Z (71)

eim = eim (de Moivres identity)

cos + i sin m = cos(m) + i sin(m) (72)mcos + i sin m

cos + i sin 1/m = cos+ 2nm

+ i sin+ 2nm

(73) 31 3 1, 1

2+ i

3

2 1

2 i

3

2


LEONHARD EULER

( )
http://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www.sut.ac.th/http://www.sut.ac.th/http://physics3.sut.ac.th/http://physics2.sut.ac.th/~suppiyahttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://physics3.sut.ac.th/


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:

15 ..1707Basel,Switzerland

: 18 ..1783St. Petersburg,Russia

http://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.html

http://en.wikipedia.org/wiki/Leonhard_Euler


ABRAHAM DE MOIVRE

( )
http://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www.sut.ac.th/http://www.sut.ac.th/http://physics3.sut.ac.th/http://physics2.sut.ac.th/~suppiyahttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.htmlhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://physics3.sut.ac.th/


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: 26 ..1667Vitry-le-Franois,Champagne,France

: 27 ..1754London,England

http://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.html

http://en.wikipedia.org/wiki/Abraham_de_Moivre


References

[1] Howard Anton, Irl C. Bivens and Stephen L. Davis,Calculus, 7th ed.,
http://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://physics2.sut.ac.th/~suppiyahttp://physics3.sut.ac.th/http://www.sut.ac.th/http://www.sut.ac.th/http://physics3.sut.ac.th/http://physics2.sut.ac.th/~suppiyahttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.htmlhttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://physics3.sut.ac.th/


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[ ] , p , , ,

Wiley, New York (2002).

[2] Eugene Hecht,Physics: Calculus, Brooks/Cole, Pacific Grove,

California (1996).

[3] David Halliday, Robert Resnick and Jearl Walker,Fundamentals of

Physics, 6th ed., Wiley, New York (2001).

[4] ,(: - ), . ().

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Institute of Plane Geometry vs. Institute of Solid Geometry

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