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eiθ

n

n

n

n

R

∇ φ(x, y, z) F(x, y, z) = Fx i + Fy j + Fz k

F(x, y, z) = Fx i + Fy j + Fz k

∇2 ∇2

(r, φ, z) (r, θ, φ)

x

dt , x > 0, t 1 loga x = (logb x)(loga b) 1 loga b = logb a ax = exp (x ln a) ln x = loge x =

sec θ = 1/ cos θ

ˆ

e = 2.718281828 . . .

cosec θ = 1/ sin θ

cot θ = 1/ tan θ

sin(−θ) = − sin θ cos(−θ) = cos θ tan(−θ) = − tan θ 2 2 2 2 2 sin θ + cos θ = sec θ − tan θ = cosec θ − cot2 θ = 1

sin(A ± B) = sin A cos B ± cos A sin B cos(A ± B) = cos A cos B ∓ sin A sin B tan A ± tan B tan(A ± B) = 1 ∓ tan A tan B 2 sin A cos B = sin(A + B) + sin(A − B )

2 cos A cos B = cos(A + B) + cos(A − B ) 2 sin A sin B = − cos(A + B) + cos(A − B ) sin A + sin B = 2 sin 21 (A + B) cos 21(A − B) sin A − sin B = 2 cos 21(A + B) sin 21 (A − B) cos A + cos B = 2 cos 21(A + B) cos 21(A − B) cos A − cos B = −2 sin 21(A + B) sin 21 (A − B)

sin 2θ = 2 sin θ cos θ cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ sin2 θ = 21 (1 − cos 2θ ) cos2 θ = 21 (1 + cos 2θ )

t = tan θ/2 sin θ =

2t 1 + t2

cos θ =

1 − t2 1 + t2

tan θ =

2t 1 − t2

A B

C

a b

c

a2 = b2 + c2 − 2bc cos A b c a = = sin C sin A sin B

cosh θ = 21 (eθ + e−θ )

sinh θ = 21(eθ − e−θ )

eθ − e−θ sinh θ = θ cosh θ e + e−θ eθ + e−θ cosh θ 1 = θ coth θ = = tanh θ e − e−θ sinh θ 1 1 sech θ = cosech θ = sinh θ cosh θ tanh θ =

cosh2 θ − sinh2 θ = 1

sech2 θ + tanh2 θ = 1 coth2 θ − cosech2 θ = 1

ln (n!) ≈ n ln n − n

for n ≫ 1

ln n! ≈ n ln n − n + 12 ln (2π )

n−1 X

m=0

(a + md) = a + (a + d) + (a + 2d) + ... + (a + (n − 1)d)

Sn =

= (n/2) [2a + (n − 1)d] = ( /2)(

n−1 X

(ar m ) = a + ar + ar 2 + ....... + ar n−1 =

m=0

|r| < 1 S∞ =

a 1−r

)

a(1 − r n ) a(r n − 1) = 1−r r−1

(1 + x)n = 1 + nx +

n(n − 1) · · · (n − r + 1) r n(n − 1) 2 x + .... + x + ... 2! r!

0! = 1 |x| < 1  x n n n (a + x) = a 1 + a

f (x)

f (x) =

x=a ∞ X 1 n=0

n!

(x−a)n f (n) (a) = f (a)+(x−a)f ′ (a)+

(x − a)3 ′′′ (x − a)2 ′′ f (a)+ · · · f (a)+ 3! 2!

  ∂f 1 ∂ 2f ∂ 2f ∂ 2f ∂f 2 2 ∆x + ∆y + ∆x + 2 ∆x∆y + 2 ∆y + · · · f (x, y) = f (x0 , y0 ) + ∂x ∂y ∂y 2! ∂x2 ∂x∂y ∆x = x − x0 , ∆y = y − y0 (x0 , y0 ).

x2 x3 + ··· + 3! 2! x2 x3 x4 − + + ··· ln (1 + x) = x − |x| < 1 3 2 4 x3 x5 x7 − + + ··· sin x = x − 5! 3! 7! x6 x2 x4 − + + ··· cos x = 1 − 4! 2! 6! x3 x5 x7 + ··· sinh x = x + + + 7! 5! 3! x6 x2 x4 + + + ··· cosh x = 1 + 4! 2! 6! 1 = 1 − x + x2 − x3 + · · · |x| < 1. 1+x 1 = 1 + x + x2 + x3 + · · · |x| < 1. 1−x

ex = 1 + x +

d d tan x = sec2 x cot x = − cosec2 x dx dx d d cosec x = − cosec x cot x sec x = sec x tan x dx dx f (x) = u(x)v(x) df dv du v =u + dx dx dx u(x)

f (u) df df du = du dx dx

df

f (x, y) df =

f (x, y)

x

∂f ∂f dx + dy ∂x ∂y

y

x(u)

y(u)

∂f dx ∂f dy df = + ∂x du ∂y du du

x dx = sin−1 2 a −x ˆ 1 x dx = tan−1 ( 2 2 a +x a a ( ˆ a+x 1 ln a−x = dx 2a = 1 ln x+a = 2 2 a −x 2a x−a ˆ

√

− cos−1

a2

√

(

)

) 1 a

x dx = sinh−1 a a2 + x2 ˆ x dx √ = cosh−1 2 2 a x −a ˆ √ √ 1 a2 − x2 dx = x a2 − x2 + 2 ˆ √ √ 1 x2 ± a2 dx = x x2 ± a2 ± 2

ˆ

x a

tanh−1 1 a

x a coth−1 ax

ln (x +

√

ln (x +

√

( |x| < a) ( |x| > a)

a2 + x2 ) x2 − a2 )

1 2 −1 x ( a sin a 2 √ 1 2 a ln (x + x2 ± a2 ) 2

)

ˆ

tan x dx

= − ln (cos x) = ln (sec x)

ˆ

cot xdx

= ln (sin x)

ˆ

sec x dx

ˆ

cosec x dx

x dx a ˆ x cos−1 dx a ˆ ax dx ˆ xn e−ax dx

  1 + sin x π  1 = ln (sec x + tan x) = ln tan + = ln 2 1 − sin x 2 4     1 1 − cos x x = ln = ln (cosec x − cot x) = ln tan 2 2 1 + cos x √ x = x sin−1 + a2 − x2 a x √ = x cos−1 − a2 − x2 a ax = ln a  n x nxn−1 n(n − 1)xn−2 −ax + ··· = −e + + a3 a2 a  n!x n! + n + n+1 (n a a a sin bx − b cos bx = eax a 2 + b2 a cos bx + b sin bx = eax a 2 + b2 sin ax x cos ax = − a a2

ˆ

sin−1

ˆ

eax sin bx dx

ˆ

eax cos bx dx

ˆ

x sin ax dx

ˆ

ln x dx

= x ln x − x

ˆ

sinh x dx

= cosh x

ˆ

tanh x dx

= ln(cosh x)

ˆ

ˆ

π 0

x



ˆ

cosh x dx = sinh x

dv u dx = uv − dx

π sin mx sin nx dx = δmn 2

ˆ

v

ˆ

du dx dx

π

cos mx cos nx dx = 0

π δmn 2

δmn δmn

∞

( 1 = 0

1 xe dx = 2 α ˆ0 ∞ 6 x3 e−αx dx = 4 α r ˆ0 ∞ 1 π 2 e−αx dx = 2 α ˆ0 ∞ 1 2 xe−αx dx = 2α 0 r ˆ ∞ 1 π 2 −αx2 xe dx = 4 α3 ˆ0 ∞ 1 2 x3 e−αx dx = 2α2 r ˆ0 ∞ 3 π 4 −αx2 xe dx = 8 α5 0 √ ˆ y π 2 erf(y ) e−x dx = 2 0 ˆ

m=n m 6= n

∞

2 α3 ˆ 0∞ n! xn e−αx dx = n+1 α r ˆ 0∞ π 2 e−αx dx = α ˆ −∞ ∞ 2 xe−αx dx = 0 −∞ r ˆ ∞ 1 π 2 −αx2 xe dx = 2 α3 −∞ ˆ ∞ 2 x3 e−αx dx = 0 −∞ r ˆ ∞ 3 π 4 −αx2 xe dx = 4 α5 −∞ r ˆ y √ 1 π 2 erf( α y) e−αx dx = 2 α 0 ˆ

−αx

x2 e−αx dx =

y0 , y1 , y2 · · · yn

ˆb a

ydx = h

y

0

2

+ y1 + y2 +

+

yn  2

y ˆb a

ydx =

h {y0 + 4(y1 + y3 + 3

f (x) = 0

+ yn−1 ) + 2(y2 + y4 +

+ yn−2 ) + yn }

xj

xj = xj−1 −

df ′ f (xj−1 ) where f = ′ dx f (xj−1 )

z = x + iy = r(cos θ + i sin θ) = reiθ z ∗ = x − iy = re−iθ p √ z |z| = x2 + y 2 = r = zz ∗ y z = tan−1 = θ x z + z∗ Re(z) = x = r cos θ = 2 z − z∗ z Im(z) = y = r sin θ = 2i z

z z

(cos θ + i sin θ)n = cos (nθ) + i sin (nθ) eiθ

e±iθ = cos θ ± i sin θ 1 cos θ = (eiθ + e−iθ ) 2 1 sin θ = (eiθ − e−iθ ) 2i eiθ − e−iθ e2iθ − 1 1 − e−2iθ = 2iθ = i tan θ = iθ e +1 e + e−iθ 1 + e−2iθ

m

Yl =



(2l + 1) (l − m)! 4π (l + m)!

1/2

l+m  1 imφ d m e (− sin θ) (cos2 θ − 1)l 2l l! d(cos θ)

Ylm (θ, φ) = Plm (cos θ)√

1 imφ e , 2π

Plm(cos θ ) 1 = √ 4π r 3 cos θ Y10 = 4π r 3 sin θe±iφ Y1±1 = ∓ 8π

Y00

Γ(n) =

ˆ

Y20

Y2±2

∞

t

n−1 −t

e dt =

0

ˆ

1 0

 5  2 cos2 θ − sin2 θ 16π r 15 =∓ cos θ sin θe±iφ 8π r 15 =∓ sin2 θe±2iφ 32π

=

Y2±1

r

 n−1 1 dt ln t

n>0 n Γ(n + 1) = nΓ(n) Γ(n)Γ(1 − n) = Γ(n + 1) = n!   √ 1 Γ = π 2

Jn (x) =

π sin nπ n

≥0 Γ(1) = 0! = 1

∞ X λ=0

 xn+2λ (−1)λ Γ(λ + 1)Γ(λ + n + 1) 2

 d  −n x Jn (x) = −x−n Jn+1 (x) dx ˆ 2π 1 exp (ix cos φ) dφ J0 (x) = 2π 0

d {xn Jn (x)} = xn Jn−1 (x) dx ˆ z 1 xJ0 (x)dx zJ1 (z) = 2π 0

   A11 A12 A13 ... A1n     A21 A22 A23 ... A2n    |A| =  A31 A32 A33 ... A3n   ... ... ... ... ...    An1 An2 An3 ... Ann  X X = (−1)k+j Akj Mkj = (−1)k+i AikMik j

Mij

Aij i

(−1)i+j Mij A

i

A

(n − 1) × (n − 1) Aij

j Aij

|A| |A| |A| |A| = 0 |A| = 0 |A| λ |A|

λ |AB| = |A| |B | n×n

a an n

n

A11 x1 + A12x2 + A13 x3 + ...... + A1n xn = 0 A21 x1 + A22x2 + A23 x3 + ...... + A2n xn = 0 ···

An1 x1 + An2 x2 + An3 x3 + ...... + Ann xn = 0

         

n

A11 A21 A31 ... An1

A12 A22 A32 ... An2

A13 A23 A33 ... An3

... ... ... ... ...

A1n A2n A3n ... Ann

     =0    

n

A11 x1 + A12x2 + A13 x3 + ...... + A1n xn + C1 = 0 A21 x1 + A22x2 + A23 x3 + ...... + A2n xn + C2 = 0 ....... An1 x1 + An2 x2 + An3 x3 + ...... + Ann xn + Cn = 0

x1        

A12 A13 A22 A23 ... ... An2 An3

−x2 =   A11 A13 ... ... C1    A21 A23 ...  ... C2    ... .... ...  .....    An1 An3 ... ... Cn 

C1 C2 .... Cn

 = ..... =       

       

A11 A21 ... An1

(−1)n A12 ... A1n A22 ... A2n ... ... ... An2 ... Ann

y1 = A11 x1 + A12x2 + A13 x3 + ...... + A1n xn y2 = A21 x1 + A22x2 + A23 x3 + ...... + A2n xn ....... yn = An1 x1 + An2 x2 + An3 x3 + ...... + Ann xn



A11 y1  A21  y2       ...  =  A31  ... yn An1 

x

y



A

A12 A22 A32 ... An2 n

A13 A23 A33 ... An3 n

... ... ... ... ...

A1n A2n A3n ... Ann



   x1   x2      ...   xn

       

A

B (A + B)ij = Aij + Bij (λA)ij = λAij X (AB)ij = Ail Blj l

AB 6= BA.

Tr(A) =

X

Aii

i

Tr(AB) = Tr(BA)

AT

A

ATij = Aji A

A†

∗ A†ij = Aji

A−1

A

AA−1 = A−1 A = I

I

A−1 (A−1 )ij = (−1)i+j Mij

(−1)j+i Mji |A| Aij

(ABC..X)−1 = X −1 ...B −1 A−1

A = AT T

A = −A

A

AT = A−1 AT A = I

±1

A = A†

n

n

A† = A−1

n×n

x

A

n

λ

Ax = λx x

A |A − λI| = 0. n n

n

B = Q−1 AQ

A TrB = TrA |B| = |A|

Q

A

Q−1 AQ

A R 3×3

R

θ

= 1

TrR = 1 + 2 cos θ u

Ru = u

Ox Oy

Oz

a · b = |a||b| cos θ (with 0 ≤ θ ≤ π) where θ is the angle between a and b.

i · i = j · j = k · k = 1. i · j = i · k = j · k = 0. a·b = 0

a

b

a

b

a = ax i + ay j + az k

b = bx i + by j + bz k

a · b = a x bx + a y by + a z bz a · a = |a|2 = ax2 + ay2 + a2z a = (a · i)i + (a · j)j + (a · k)k

n a × b = |a||b| sin θˆ ˆn ˆn

0≤θ≤π

θ

i×i= j×j = k×k = 0 i × j = k,

j × k = i,

k×i= j

a × b = −b × a a = ax i + ay j + az k

b = bx i + by j + bz k

   a × b =  

 i j k  ax ay az  bx by bz 

a × b = |a||b| sin θ θ a×b = 0

[a b c] = a · (b × c) which also equals (a × b) · c [a b c] = [b c a] = [c a b] = − [a c b] = − [b a c] = − [c b a] a b

[a b c] = 0.

c

a b

c

[a b c]

a × (b × c) = (a · c)b − (a · b)c a × (b × c) 6= (a × b) × c ∇ ∇=i

∂ ∂ ∂ +j +k . ∂z ∂x ∂y φ(x, y, z)

grad φ = ∇φ = i

∂φ ∂φ ∂φ +j +k . ∂z ∂x ∂y

∇φ φ F(x, y, z) = Fx i + Fy j + Fz k ∂Fx ∂Fy ∂Fz + + ∂z ∂y ∂x ∂F ∂F ∂F +j· +k· div F = i · ∂z ∂x ∂y

div F = ∇ · F =

F(x, y, z) = Fx i + Fy j + Fz k

curl F = ∇ × F = i



∂Fz ∂Fy − ∂z ∂y



+j



∂Fx ∂Fz − ∂x ∂z



+k



∂Fy ∂Fx − ∂y ∂x

  i j k ∂F ∂F  ∂ ∂F ∂ ∂ +j× +k× =  ∂x ∂y curl F = i × ∂z ∂x ∂y ∂z  Fx Fy Fz

div grad φ = ∇ · (∇φ) = ∇2 φ =

     

∂ 2φ ∂ 2φ ∂ 2φ + + ∂x2 ∂y 2 ∂z 2

∇2

div curl F = ∇ · (∇ × F) = [ ∇ ∇ F] = 0. curl grad φ = ∇ × (∇φ) = 0 curl curl F = ∇ × (∇ × F) = ∇(∇ · F) − ∇2 F = grad divF − ∇2 F ∇2 F =

∂ 2F ∂ 2F ∂ 2F + 2 + 2 ∂x2 ∂y ∂z ∇

∇(φ + ψ) = ∇φ + ∇ψ ∇ · (a + b) = ∇ · a + ∇ · b ∇ × (a + b) = ∇ × a + ∇ × b ∇(φ ψ) = φ ∇ψ + ψ ∇φ ∇(a · b) = (b · ∇)a + (a · ∇)b + b × (∇ × a) + a × (∇ × b) ∇ · (φ a) = φ ∇ · a + (∇φ) a



.

∇ · (a × b) = b · ∇ × a − a · ∇ × b ∇ × (φ a) = φ ∇ × a + (∇φ) × a ∇ × (a × b) = (b · ∇)a − (a · ∇)b + a (∇ · b) − b (∇ · a)

V

S F(x, y, z) ‹ dS = ndS ˆ

S

F · dS =

˚

V

∇ · F dV

dS

S

C

n F(x, y, z) ˛ dr = i dx + j dy + k dz

C

F · dr =

¨

S

∇ × F · dS

dS = ndS ˆ

x = r cos φ, y = r sin φ, z = z r ≥ 0, 0 ≤ φ ≤ 2π, −∞ ≤ z ≤ ∞ r=

p

(x2 + y 2 ),

φ = tan−1 (y/x) ,

z=z

z=0

x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ r ≥ 0, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π r= ∇2

p

(x2 + y 2 + z 2 ),

1∂ ∇ f= r ∂r 2

φ = tan−1 (y/x) ,

(r, φ, z)   ∂f 1 ∂ 2f ∂ 2f r + 2 2+ 2 ∂r ∂z r ∂φ

θ = cos−1 (z/r )

∇2

(r, θ, φ) 1 ∂ ∇ f= 2 r ∂r 2

    ∂ 2f 1 ∂f 1 ∂ 2 ∂f + 2 2 sin θ + 2 r ∂θ ∂r r sin θ ∂θ r sin θ ∂φ2

ds = |dr|

dV

p ds = (dr )2 + r 2 (dφ)2 + (dz )2 dV = r dr dφ dz

ds =

q

(dr )2 + r 2 (dθ)2 + r 2 sin2 θ(dφ)2

dV = r 2 sin θ dr dθ dφ

f (t)

t

∞ X

f (t) =

T

f (t + T ) = f (t)

Fn e−iωn t ,

n=−∞

where ωn = and Fn =

0

T

1 T

−T /2 1 T

2πn (n = 0, ±1, ±2.....∞) ˆT eiωn t f (t) dt

T

T /2 ˆ

e−iωn t eiωm t dt = δnm T

δnm f (t) f (t) = a0 +

∞ X

an cos ωn t + bn sin ωn t

n=1

ˆ 2 f (t) cos ωn t dt where an = T T ˆ 2 f (t) sin ωn t dt bn = T T ˆ 1 f (t) dt and a0 = T T

T → ∞ f (t) 1 f (t) = 2π

ˆ

∞

F (ω)e−iωt dω −∞ ˆ ∞ f (t)eiωt dt where F (ω) = −∞

f (t)

F (ω) ˆ

∞ ′

−∞

e−iωt eiω t dt = 2πδ(ω − ω ′ )

f (t)

f (t − a)

a

F (ω ) is replaced by F (ω )eiωa ′

eiω t

f (t)

F (ω ) is ‘translated′ into F (ω + ω ′ )

f (t)

g(t)

t h(t)

h(t) = f (t) ∗ g(t) = =

ˆ

∞

ˆ −∞ ∞

−∞

f (u)g(t − u)du f (t − u)g(u)du

h(t) H(ω) = F (ω)G(ω) f (t) g(t) H(ω ) = F (ω) ∗ G(ω )

1 f (t) = 2π

ˆ

F (ω )e

−∞

f (t) = e−iω0 t f (t) = sin ω0 t f (t) = cos ω0 t f (t) = δ(t − t0 )

G(ω)

h(t) = f (t)g(t)

∞ −iωt

F (ω)

dω

F (ω ) =

ˆ

∞

f (t)eiωt dt

−∞

F (ω) = 2πδ (ω − ω0 ) π F (ω) = [δ (ω + ω0 ) − δ (ω − ω0 )] i F (ω) = π [δ(ω + ω0 ) + δ(ω − ω0 )] F (ω) = eiωt0

f (t)



= T1 (|t| < T /2) = 0 (|t| ≥ T /2)

f (t) =

√1 2πτ

f (t) =

1 2τ

 2  exp − 2τt 2

  exp − τ|t|

F (ω) =

sin (ωT /2) ωT /2

  F (ω) = exp −ω 2 τ 2 /2

F (ν) = 1/(1 + ω 2 τ 2 )

e2πiSr cos φ ˆ 2π ˆ

a

φ=0 r=0

e2πiSr cos φ rdrdφ =

aJ1 (2πSa) S

J1 (x) = when |x| = 1.22π(= 3.3833), 2.233π (= 7.016), 3.238π(= 10.174), ... x = max. when |x| = 0, 2.679π(= 8.417), ... = min. when |x| = 1.635π(= 5.136), 3.699π (= 11.620), ...

F (s)

f (t) F (s) =

ˆ

∞

f (t)e−st dt

0

Function f (t) c1 f1 (t) + c2 f2 (t) f (at) eat f (t) f (t) =



(t − a) t > a 0 t...