Math Physics
description
Transcript of Math Physics
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5/20/2018 Math Physics
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I(Basic Mathematics for Physics I)
Suppiya Siranan
.http://physics2.sut.ac.th/~suppiya/note/Mathematics01.pdf
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 1/75
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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/
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 2/75
http://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://nobelprize.org/nobel_prizes/physics/laureates/1933/http://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://en.wikipedia.org/wiki/Paul_Dirachttp://nobelprize.org/nobel_prizes/physics/laureates/1933/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://nobelprize.org/nobel_prizes/physics/laureates/1933/http://en.wikipedia.org/wiki/Paul_Dirachttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://nobelprize.org/nobel_prizes/physics/laureates/1933/http://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.htmlhttp://en.wikipedia.org/wiki/Paul_Dirachttp://en.wikipedia.org/wiki/Paul_Dirachttp://physics3.sut.ac.th/http://physics3.sut.ac.th/ -
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(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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 3/75
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(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
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I
(Factorial & Binomial Theorem)
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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
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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
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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
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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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 9/75
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: 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 . . .
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 10/75
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II
(Summation Formulae)
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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)
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(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
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(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)
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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
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III
(Exponential & LogarithmFunctions)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 16/75
E ti l F ti
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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
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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
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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
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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+
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IV
(Trigonometry)
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Trigonometry
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 22/75
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(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
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V
(Differential Calculus)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 25/75
Derivative
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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)
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 27/75
3. f(x) g(x)
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( ) ( )
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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 28/75
4. f(x)/g(x) g(x) = 0
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 29/75
5. fg(x) f(z) z g(x)
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 30/75
I
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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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 31/75
II
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ddx
sin1 x= 11 x2 (|x| 1)
ddx
cot1 x= 11 +x2
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 32/75
Higher Order Derivatives
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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), . . . !!
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 33/75
n
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 34/75
GOTTFRIED WILHELM VON LEIBNIZ ( LEIBNITZ)
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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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 35/75
Partial Derivatives
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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
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 37/75
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VI
(Integral Calculus)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 38/75
Indefinite Integral (Antiderivative)
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 39/75
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0 dx=C
xn dx=
xn+1
n+ 1+C
( n = 1)
x1 dx= ln |x| +C
ex dx= ex +C
ax dx=
ax
ln a
+C
sin xdx= cos x+C cos xdx= sin x+C
tan xdx= ln |sec x| +C
sec xdx= ln |sec x+ tan x| +C csc xdx= ln |csc x cot x| +C
cot xdx= ln |sin x| +C
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 40/75
Definite Integral
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(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)
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 42/75
GEORG FRIEDRICH BERNHARD RIEMANN
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( )
: 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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 43/75
: 31
x2 dx
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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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 44/75
3x2 dx = 2 lim
1 + 2
n(n+ 1)+
2 n(n+ 1)(2n+ 1)
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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
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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
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(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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 47/75
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VII
(Taylor & Maclaurin Series)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 48/75
Taylor & Maclaurin Series
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(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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 49/75
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/ -
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( )
: 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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 50/75
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/ -
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( )
: ..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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 51/75
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/ -
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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)
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(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
-
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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
-
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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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 55/75
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VIII
(Scalar & Vector)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 56/75
Scalar & Vector
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(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
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1. A B
A=
B
2. (zero vector) 0
0= 03. Am mA mAA
(a) m >0 A = 0 m
A
A
(b) m
-
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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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 59/75
Scalar Product (Dot Product)
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(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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 60/75
Vector Product (Cross Product)
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(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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 61/75
= =kk= 0 = =k, k= k= k = k=
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 62/75
Gradient, Divergence & Curl
( fi ld) A() A( )
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(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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 63/75
(divergence) A= A(r)
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 64/75
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IX
(Complex Number)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 65/75
Complex Number
2 + 1 0 R
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 66/75
2 a+ ibc+ id
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(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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 67/75
(modulus) z |z|(|z| 0)
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|z| x2 +y2 = (x+ iy)(x iy) (67)|z| = zz = zz
z =x+ iy (x, y)(xy)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 68/75
z =x+ iy (r, ) x r cos y r sin
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 69/75
Euler Formula
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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)
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 70/75
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(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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 71/75
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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 72/75
ABRAHAM DE MOIVRE
( )
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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
Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 73/75
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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Suppiya Siranan (PhysicsSUT) Basic Mathematics for Physics I p. 75/75
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