Bernoulli Trials
P
(
E
)
=
n
k
p
k
(
1
-
p
)
n
-
k
Cauchy-Schwarz Inequality
k
=
1
n
a
k
b
k
2
≤
k
=
1
n
a
k
2
k
=
1
n
b
k
2
Cauchy Formula
f
(
z
)
·
Ind
γ
(
z
)
=
1
2
π
i
γ
f
(
ξ
)
ξ
-
z
d
ξ
Cross Product
V
1
×
V
2
=
i
j
k
∂
X
∂
u
∂
Y
∂
u
0
∂
X
∂
v
∂
Y
∂
v
0
Vandermonde Determinant
1
1
⋯
1
v
1
v
2
⋯
v
n
v
1
2
v
2
2
⋯
v
n
2
⋮
⋮
⋱
⋮
v
1
n
-
1
v
2
n
-
1
⋯
v
n
n
-
1
=
1
≤
i
<
j
≤
n
(
v
j
-
v
i
)
Lorenz Equations
x
=
σ
(
y
-
x
)
y
=
ρ
x
-
y
-
x
z
z
=
-
β
z
+
x
y
Maxwell's Equations
∇
×
B
-
1
c
∂
E
∂
t
=
4
π
c
j
∇
·
E
=
4
π
ρ
∇
×
E
+
1
c
∂
B
∂
t
=
0
∇
·
B
=
0
Einstein Field Equations
R
μ
ν
-
1
2
g
μ
ν
R
=
8
π
G
c
4
T
μ
ν
Ramanujan Identity
1
(
φ
5
-
φ
)
e
25
π
=
1
+
e
-
2
π
1
+
e
-
4
π
1
+
e
-
6
π
1
+
e
-
8
π
1
+
…
Another Ramanujan identity
k
=
1
∞
1
2
⌊
k
·
φ
⌋
=
1
2
0
+
1
2
1
+
⋯
Rogers-Ramanujan Identity
1
+
k
=
1
∞
q
k
2
+
k
(
1
-
q
)
(
1
-
q
2
)
⋯
(
1
-
q
k
)
=
j
=
0
∞
1
(
1
-
q
5
j
+
2
)
(
1
-
q
5
j
+
3
)
,
f
o
r
|
q
|
<
1
.
Commutative Diagram
H
←
K
↓
↑
H
→
K