MIDTERM FORMULA SHEET
SOME INTEGRALS OFTEN MET IN EM PROBLEMS
1
∫ x dx = ln | x | +C
1
x
∫ (a 2 ± x 2 )3 / 2 dx = ± a 2 a 2 ± x 2 + C
x
1
∫ (a 2 + x 2 )3 / 2 dx = − a 2 + x 2 + C

VECTOR IDENTITIES
G G G
G G G
G G G
A ⋅ ( B × C ) = C ⋅ ( A × B ) = B ⋅ (C × A)
G G G
G G G
G G G
A × ( B × C ) = B( A ⋅ C ) − C ( A ⋅ B)

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om

∇(Φ + Ψ ) = ∇Φ + ∇Ψ
G G
G
G
∇ ⋅ ( A + B) = ∇ ⋅ A + ∇ ⋅ B
G G
G

G
∇ × ( A + B) = ∇ × A + ∇ × B
∇(ΦΨ ) = Φ∇Ψ + Ψ∇Φ
 Φ  Ψ∇Φ − Φ∇Ψ
∇  =
Ψ2
Ψ

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co
an
th

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∇Φ n = nΦ n−1∇Φ
G
G
G
∇ ⋅ (ΦA) = A ⋅∇Φ + Φ∇ ⋅ A
G G
G G
G
G
∇ ⋅ ( A × B ) = B ⋅∇ × A − A ⋅∇ × B
G
G
G
∇ × (ΦA) = ∇Φ × A + Φ∇ × A

G G
G G G G G
G G
G
∇ × ( A × B ) = A∇ ⋅ B − B∇ ⋅ A + ( B ⋅∇) A − ( A ⋅∇) B
G G
G
G
G
G
G
G
G G
∇( A ⋅ B ) = A × (∇ × B ) + B × (∇ × A) + ( A ⋅∇) B + ( B ⋅∇) A

cu

u

du
o

∇ ⋅∇Φ = ∇ 2Φ
G
∇ ⋅∇ × A = 0
∇ × ∇Φ = 0
G
G
G
∇ ×∇ × A = ∇∇ ⋅ A − ∇ 2 A


VECTOR INTEGRAL THEOREMS
G
G G
∫∫∫ (∇ ⋅ A)dv = w
∫∫ A ⋅ ds (Divergence theorem, Gauss identity)
v

S[ v ]

G G
∫∫ (∇ × A) ⋅ ds =
S

v∫

G G
A ⋅ dl (Curl theorem 1, Stokes’ theorem)

C[ S ]

G
G
G G
ˆ
∇
×
A
dv
=

ds
×
A
≡
n
×
A
(
)
(
)ds (Curl theorem 2)
∫∫∫
w
∫∫
w
∫∫
v

S[ v ]

(

(

)

S[ v ]

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)

x2
x
2
2
∫ (a 2 + x 2 )3 / 2 dx = − a 2 + x 2 + ln x + a + x + C
1
1
x
∫ a 2 + x 2 dx = a arctan a + C
1
 1 a−x
 x
 2a ln  a + x  = − a arctanh  a  ,| x |< a
1



 
∫ x 2 − a 2 dx =  1
 − arccoth  x 
,| x |> a
 
 a
a
x
1
2
2

∫ a 2 + x 2 dx = 2 ln a + x + C
x
2
2
∫ a 2 + x 2 dx = a + x + C
1
2
2
∫ a 2 + x 2 dx = ln( x + a + x ) + C
1
1  a + a 2 + x2 
+C
ln 
dx
=
−
∫ x a2 + x2

a 
x


1
∫ x sin(ax)dx = a 2 [sin(ax) − ax cos(ax)]
1
∫ x cos(ax)dx = a 2 [cos(ax) + ax sin(ax)]

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MIDTERM FORMULA SHEET

 x ag − bf
1

=
arctan
∫ (ax 2 + b) fx 2 + g b ag − bf
 b fx 2 + g

dx


, (ag > bf )



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om

 x

Rectangular ↔ Spherical

∫ sin x dx = ln tan  2  + C
1

x = R sin θ cos φ
y = R sin θ sin φ
z = R cos θ

x π


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∫ cos x dx = ln tan  2 + 4  + C
SOME USEFUL DEFINITE INTGERALS
2π
0 , m ≠ n
∫ sin mx ⋅ sin nx dx = π , m = n ≠ 0
0

∫

sin mx ⋅ cos nx dx = 0
, m≠n
m=n≠0

0

, m≠n
m=n≠0

∫ cos mx ⋅ cos nx dx = π / 2,
0

,
0

∫ sin mx ⋅ cos nx dx =  2m ,
 m 2 − n 2
0

π
π
(a − b cos x)
 ,
∫ (a 2 + b2 − 2ab cos x) dx =  a
 0,
0
π

m + n = even number

cu

0

π

u

0

∫ sin mx ⋅ sin nx dx = π / 2,

du
o

0

π


co

, m≠n
m=n≠0

an

0
2π

r = R sin θ
φ =φ
z = R cos θ

th

0

∫ cos mx ⋅ cos nx dx =  π ,

m + n = odd number

a>b>0
b>a>0

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R = x2 + y 2 + z 2

(


θ = arccos z / x 2 + y 2 + z 2
φ = arctan( y / x)

)

Cylindrical ↔ Spherical

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2π

r = x2 + y 2
 y
φ = arctan  
 x
z=z

x = r cos φ
y = r sin φ
z=z

π

∫ tan xdx = − ln | cos x | +C , x ≠ (2k + 1) 2
∫ cot xdx = ln | sin x | +C , x ≠ 2kπ
1

COORDINATE TRANSFORMATIONS
Rectangular ↔ Cylindrical


R = r2 + z2
φ =φ
θ = arccos z / r 2 + z 2

(

)

VECTOR TRANSFORMATIONS
Rectangular Components ↔ Cylindrical Components
ax = ar cos φ − aφ sin φ
ar = ax cos φ + a y sin φ
a y = ar sin φ + aφ cos φ
aφ = − ax sin φ + a y cos φ
a z = az
az = az

Note: φ is the position angle of the point at which the vector exists.
Rectangular Components ↔ Spherical Components
ax = aR sin θ cos φ + aθ cosθ cos φ − aφ sin φ
a y = aR sin θ sin φ + aθ cos θ sin φ + aφ cos φ
az = aR cos θ − aθ sin θ
aR = ax sin θ cos φ + a y sin θ sin φ + az cos θ
aθ = ax cos θ cos φ + a y cos θ sin φ − a z cos θ
aφ = − ax sin φ + a y cos φ

Note: φ and θ are the position angles of the point at which the
vector exists.


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MIDTERM FORMULA SHEET

cu

u

du
o

ng

th

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om

an

DERIVATIVES OF ELEMENTARY FUNCTIONS
(const.)′ = 0
1
(arctan x)′ =
( x)′ = 1
1 + x2
1
( x k )′ = kx k −1
(arc cot x)′ = −
1 + x2

(e x )′ = e x
(sinh x)′ = cosh x
(a x )′ = a x ln a
(cosh x)′ = sinh x
1
1
(ln x)′ =
(tanh x)′ =
= 1 − tanh 2 x
x
cosh 2 x
1
(log a x)′ =
, a ≠ 1, x > 0
1
(coth x)′ = −
= 1 − coth 2 x
x ln a
2
sinh
x
(sin x)′ = cos x
1
(cos x)′ = − sin x
(arcsinh x)′ =
1 + x2
1
(tan x)′ =
, x ≠ (2k + 1)π
1

cos 2 x
(arccosh x)′ = ±
, x >1
x2 − 1
1
(cot x)′ = − 2 , x ≠ kπ
1
sin x
(arctanh x)′ =
,| x |< 1
1
1 − x2
(arcsin x)′ =
,| x |< 1
1
1 − x2
(arccoth x)′ =
,| x |> 1
2
1
−
x
1
(arccos x)′ = −
,| x |< 1
1 − x2

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Note: θ is the position angle of the point at which the vector exists.


G ∂F ∂Fy ∂Fz
∇⋅F = x +
+
∂x
∂y
∂z
G
 ∂F ∂Fy 
 ∂Fx ∂Fz   ∂Fy ∂Fx 
∇ × F = xˆ  z −
−
−
 + yˆ 

 + zˆ 
∂z 
∂x   ∂x
∂y 
 ∂z
 ∂y
∂ 2Φ ∂ 2Φ ∂ 2Φ
∇ ⋅ (∇Φ ) ≡ ∇ 2 Φ ≡ ∆Φ = 2 + 2 + 2
∂x
∂y
∂z
G
2
2
2

2
∇ F = xˆ∇ Fx + yˆ∇ Fy + zˆ∇ Fz
Cylindrical Coordinates
∂Φ
1 ∂Φ
∂Φ
ϕˆ +
∇Φ =
rˆ +
zˆ
r ∂ϕ
∂r
∂z
G 1 ∂
1 ∂Fϕ ∂Fz
(rFr ) +
∇⋅F =
+
r ∂r
r ∂ϕ
∂z
G
 1 ∂Fz ∂Fφ  ˆ  ∂Fr ∂Fz   1 ∂ (rFφ ) 1 ∂Fr 
∇ × F = rˆ 
−
−
−
 +φ 

 + zˆ 

∂z 
∂r   r ∂r
r ∂φ 
 ∂z
 r ∂φ
1 ∂  ∂Φ  1 ∂ 2 Φ ∂ 2 Φ
∇ ⋅ (∇Φ ) ≡ ∇ 2 Φ ≡ ∆Φ =
+
r
+
r ∂r  ∂r  r 2 ∂φ 2 ∂z 2
G
 ∂ 2 A 1 ∂Ar Ar 1 ∂ 2 Ar 2 ∂Aφ ∂ 2 Ar 
∇ 2 A = rˆ  2r +
−
+
−
+ 2 +
r ∂r r 2 r 2 ∂φ 2 r 2 ∂φ
∂z 
 ∂r
 ∂ 2 Aφ 1 ∂Aφ Aφ 1 ∂ 2 Aφ 2 ∂Ar ∂ 2 Aφ 
ˆ
φ 2 +
− 2 + 2
+ 2
+
+
2
2 

 ∂r
r
r
φ
∂
∂
r
r
r
z
φ
∂
∂


2
2
2
 ∂ A 1 ∂Az 1 ∂ Az ∂ Az 
zˆ  2z +
+ 2
+ 2 
2
r
r
∂
dr
r
φ
∂

∂z 

Spherical Coordinates
∂Φ ˆ 1 ∂Φ ˆ
1 ∂Φ
θ+
ϕˆ
R+
∇Φ =
R ∂θ
R sin θ ∂ϕ
∂R
G 1 ∂
1
1 ∂Fϕ
∂
( R 2 FR ) +
( Fθ sin θ ) +
∇⋅F = 2
R sin θ ∂θ
R sin θ ∂ϕ
R ∂R

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Cylindrical Components↔ Spherical Components
ar = aR sin θ + aθ cos θ
aR = ar sin θ + az cos θ
aφ = aφ
aθ = ar cos θ − az sin θ

aφ = aφ
a z = aR cos θ − aθ sin θ

DIFFERENTIAL OPERATORS
Rectangular Coordinates
∂Φ
∂Φ
∂Φ
∇Φ = xˆ
+ yˆ
+ zˆ
∂x
∂y
∂z

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MIDTERM FORMULA SHEET
 ∂
∂Aθ
 ∂θ ( Aϕ sin θ ) − ∂ϕ


1  1 ∂AR ∂
θˆ 
( RAϕ )  +
−
R  sin θ ∂ϕ ∂R



G
ˆ + θˆRdθ + ϕˆ R sin θ dϕ ;
dl = RdR
G ˆ 2
ds = RR
sin θ dθ dϕ + θˆR sin θ dRdϕ + ϕˆ RdRdθ ;


+


dv = R 2 sin θ dRdθ dϕ

1 ∂
∂A 
( RAθ ) − R 

R  ∂R
∂θ 

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∂ 2Φ
∂ϕ 2


+



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th

 ∂ 2 Aθ 2 ∂Aθ
Aθ
1 ∂ 2 Aθ cot θ ∂Aθ
ˆ
−
+
+ 2
+
θ 2 +
R ∂R R 2 sin 2 θ R 2 ∂θ 2
R ∂θ
 ∂R
∂ 2 Aθ
1
2 ∂AR 2 cot θ ∂Aϕ 
+
−
+
R 2 sin 2 θ ∂ϕ 2 R 2 ∂θ R 2 sin θ ∂ϕ 

co

1 ∂  2 ∂Φ 
1
∂ 

∂Φ 
1
R
+ 2
 sin θ
+ 2 2
2
∂R  R sin θ ∂θ 
∂θ  R sin θ
R ∂R 
2
2
G
 ∂ AR 2 ∂AR 2
1 ∂ AR cot θ ∂AR
∇ 2 A = Rˆ 
+
− 2 AR + 2
+ 2
+
2
R ∂R R
R ∂θ 2
R ∂θ
 ∂R
∂Aϕ
∂ 2 AR 2 ∂Aθ 2 cot θ
1
2
−

−
−
A
θ
R 2 sin 2 θ ∂ϕ 2 R 2 ∂θ
R2
R 2 sin θ ∂ϕ

∇ 2Φ =

ELECTROMAGNETIC EQUATIONS
Coaxial line
2πε
µ
b µ
, F/m; L1 = 0 ln   + 0 , H/m
C1 =
ln(b / a)
2π  a  8π
Twin-lead line
2

πε
µ  h
h
C1 =
F/m; L1 = ln +   − 1  H/m
2
π r
h



r
h




ln +   − 1
r

r



an

ϕˆ

1
R sin θ

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om

G
∇ × A = Rˆ

du
o


2
 ∂ 2 Aϕ 2 ∂Aϕ
Aϕ
1 ∂ Aϕ cot θ ∂Aϕ
+
−
+
+ 2
+
ϕˆ 
 ∂R 2 R ∂R R 2 sin 2 θ R 2 ∂θ 2
∂
θ
R


cu

u

∂ 2 Aϕ
∂AR 2 cot θ ∂Aθ 
1
2
+
+

R 2 sin 2 θ ∂ϕ 2 R 2 sin θ ∂ϕ R 2 sin θ ∂ϕ 


DIFFERENTIAL ELEMENTS
Cartesian coordinates
G
G
ˆ + ydy
ˆ + zdz
ˆ
ˆ
ˆ ; ds = xdydz
ˆ
dl = xdx
+ ydxdz
+ zdxdy
; dv = dxdydz
Cylindrical coordinates
G
G
ˆ + ϕˆ rdϕ + zdz
ˆ ϕ dz + ϕˆ drdz + zrdrd
ˆ ; ds = rrd
ˆ
dl = rdr
ϕ ; dv = rdrdϕ dz
Spherical coordinates

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