Download Geometry and trigonometry

• Geometry and trigonometry
Circle of radius r: circumference = 2πr; area = πr2.
Sphere of radius r: area = 4πr2; volume = 4/3πr3.
Right circular cylinder of radius r and height h:
Area = 2πr2+2πrh; volume = πr2h.
Triangle of base b and altitude h: area = 1/2 bh.
Quadratic Formula
− b × b 2 − 4ac
If ax + bx + c = 0, then x =
2a
2
Trigonometric Functions of Angle θ
sinθ = y/r
tanθ = y/x
secθ = r/x
cosθ = x/r
cotθ = x/y
cscθ = r/y
Trigonometric Identities
sin(
π
2
− θ ) = cos θ
π
− θ ) = sin θ
2
sin θ / cos θ = tan θ
cos(
sin 2 θ + cos 2 θ = 1
sec 2 θ − tan 2 θ = 1
csc 2 θ − cot 2 θ = 1
sin 2θ = 2 sin θ cos θ
cos 2θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ
sin (α ± β) = sin α cos β ± cos α sin β
cos (α ± β) = cos α cos β ∓ sin α sin β
tan α ± tan β
tan (α ± β) =
1 ∓ tan α tan β
1
1
sin α ± sin β = 2 sin (α ± β) cos (α ± β)
2
2
• Expansions
Binomial:
nx n(n − 1) x 2
+
± ( x 2 < 1)
1!
2!
nx n(n + 1) x 2
= 1∓
+
∓ ( x 2 < 1)
1!
2!
(1 ± x) n = 1 ±
(1 ± x) − n
Exponential:
x2 x3
+
2! 3!
x2 1 3
ln(1 ± x) = x −
+ x −
2! 3!
ex = 1+ x +
Trigonometric:
sin θ = θ −
cos θ = 1 −
θ3
3!
θ2
+
5!
θ4
−
−
4!
θ 3 2θ 5
+
+
tan θ = θ +
3! 15
2!
+
θ5
• Derivative and Integrals
dx
=1
dx
d
du
(au ) = a
dx
dx
d
du dv
(u + v) =
+
dx
dx dx
d m
x = mx m −1
dx
d
1
ln x =
dx
x
d
du
dv
(uv) = v
+u
dx
dx
dx
d x
e = ex
dx
d
sin x = cos x
dx
d
cos x = − sin x
dx
d
tan x = sec 2 x
dx
d
cot x = − csc 2 x
dx
d
sec x = tan x sec x
dx
d
csc x = − cot x csc x
dx
d u
du
e = eu
dx
dx
d
du
sin u = cos u
dx
dx
d
du
cos u = − sin u
dx
dx
∫ dx = x
∫ (au)dx = a ∫ udx
∫ (u + v)dx = ∫ udx + ∫ vdx
m
∫ x dx =
mx m −1
(m ≠ 1)
m +1
1
∫ x = ln x
dv
du
∫ u dx dx = uv − ∫ v dx dx
∫ e dx = e
x
x
∫ sin xdx = − cos x
∫ cos xdx = sin x
∫ tan xdx = − ln cos x
1
1
x − sin 2 x
2
4
1
1
2
∫ cos xdx = 2 x + 4 sin 2 x
1 − ax
− ax
∫ e dx = − a e
1
− ax
− ax
∫ xe dx = − a 2 (ax + 1)e
1 2 2
2 − ax
− ax
∫ x e dx = − a 3 (a x + 2ax + 2)e
n!
n − ax
∫ x e dx = a n+1
∞
1 ⋅ 3 ⋅ 5 (2n − 1) π
2 n − ax
∫0 x e = 2 n+1 a n
a
∫ sin
∫
2
xdx =
dx
(x 2 ± a 2 )3
=
±x
a2 x2 ± a2
•
Vector Derivatives
Cartesian.
∂t
∂t
∂t
xˆ +
yˆ +
zˆ
∂x
∂y
∂z
∂ v ∂ v y ∂ vz
Divergence: ∇ ⋅ v = x +
+
∂x ∂y ∂z
∂ v y ∂ vx
∂v ∂v
∂ v ∂ vy
) xˆ + ( x − z ) yˆ + (
−
) zˆ
Curl: ∇ × v = ( z −
∂y ∂z
∂z ∂x
∂x ∂y
Gradient: ∇t =
Laplacian: ∇ 2 t =
∂ 2t ∂ 2t ∂ 2 t
+
+
∂ x2 ∂ y2 ∂ z2
Spherical.
∂t
1 ∂t ˆ
1 ∂t
rˆ +
θ+
φˆ
∂r
r ∂θ
r sin θ ∂ φ
∂
1 ∂
1
1 ∂ vφ
( r 2 v r )+
Divergence: ∇ ⋅ v = 2
(sin θvθ )+
r sin θ ∂ θ
r sin θ ∂ φ
r ∂r
Gradient: ∇t =
Curl:
∇×v =
∂ (sin θvφ ) ∂ vθ
1
1 1 ∂ v r ∂ (rvφ ) ˆ 1 ∂ (rvθ ) ∂ v r
]θ + [
]φˆ
−
−
−
[
]rˆ+ [
∂θ
∂φ
∂r
∂θ
r sin θ
r sin θ ∂ φ
r ∂r
Laplacian: ∇ 2 t =
Cylindrical.
∂t
∂ 2t
1 ∂ 2 ∂t
1
∂
1
(
)
+
(sin
θ
)
r
+
∂ r r 2 sin θ ∂θ
∂ θ r 2 sin 2 θ ∂ φ 2
r 2 ∂r
∂t
∂t
1 ∂t
ρˆ +
φˆ +
zˆ
∂ρ
ρ ∂φ
∂z
1 ∂
1 ∂ vφ ∂ v z
Divergence: ∇ ⋅ v =
( ρv ρ ) +
+
ρ ∂ρ
ρ ∂φ
∂z
∂ vρ ∂ vz
1 ∂ ( ρvφ ) ∂ v ρ
1 ∂ v z ∂ vφ
Curl: ∇ × v = [
−
−
−
]zˆ
] ρˆ + [
]φˆ + [
ρ ∂φ ∂ z
∂z ∂s
ρ ∂ρ
∂θ
∂t
1 ∂
1 ∂ 2t ∂ 2 t
2
Laplacian: ∇ t =
(ρ
)+
+
ρ ∂ρ ∂ ρ ρ 2 ∂φ 2 ∂ z 2
Gradient: ∇t =
•
Spherical and cylindrical coordinates
Spherical
⎧ x = r sin θ cos φ
⎪
⎨ y = r sin θ sin φ
⎪ z = r cos θ
⎩
⎧ xˆ = sin θ cos φ rˆ + cos θ cos φ θˆ − sin φ φˆ
⎪⎪
⎨ yˆ = sin θ sin φ rˆ + cos θ sin φ θˆ + cos φ φˆ
⎪
⎪⎩ zˆ = cos θ rˆ − sin θ θˆ
⎧r = x 2 + y 2 + z 2
⎪
⎪
−1
2
2
⎨θ = tan ( x + y / z )
⎪
−1
⎪⎩φ = tan ( y / x)
⎧ rˆ = sin θ cos φxˆ + sin θ sin φyˆ + cos θzˆ
⎪ˆ
⎨θ = cos θ cos φxˆ + cos θ sin φyˆ − sin θzˆ
⎪φˆ = − sin φxˆ + cos φzˆ
⎩
Cylindrical
⎧ x = ρ cos φ
⎪
⎨ y = ρ sin φ
⎪z = z
⎩
⎧ xˆ = cos φ ρˆ − sin φ φˆ
⎪
⎨ yˆ = sin φ ρˆ + cos φ φˆ
⎪ zˆ = zˆ
⎩
⎧ρ = x 2 + y 2
⎪⎪
−1
⎨φ = tan ( y / x)
⎪z = z
⎪⎩
⎧ ρˆ = cos φxˆ + sin φyˆ
⎪
⎨ φˆ = − sin φxˆ + cos φyˆ
⎪ zˆ = zˆ
⎩
•
Vector Identities
Triple Products
1. A ⋅ (B × C) = B ⋅ (C × A) = C ⋅ (A × B)
2. A × (B × C) = B(C ⋅ A) − C ( A ⋅ B)
Product Rules
1.
2.
3.
4.
5.
6.
∇( fg ) = f (∇g ) + g (∇f )
∇(A ⋅ C) = A × ( ∇ × B) + B × ( ∇ × A) + (A ⋅ ∇ )B + (B ⋅ ∇ )A
∇ ⋅ ( fA) = f ( ∇ ⋅ A) + A(∇ ⋅ f )
∇ ⋅ (A × B) = B ⋅ ( ∇ × A) − A ⋅ ( ∇ × B)
∇ × ( fA) = f ( ∇ × A) − A × (∇f )
∇ × (A × B) = (B ⋅ ∇ )A − (A ⋅ ∇ )B + A( ∇ ⋅ B) − B( ∇ ⋅ A)
Second Derivatives
1. ∇ ⋅ ( ∇ × A) = 0
2. ∇ × (∇f ) = 0
3. ∇ × ( ∇ × A) = ∇( ∇ ⋅ A) − ∇ 2 A
•
Fundamental Theorems:
Gradient Theorem:
∫
b
a
(∇ f ) ⋅ dl = f (b) − f (a )
Divergence Theorem: ∫ (∇ ⋅ A)dτ = ∫ A ⋅ da
Curl Theorem: ∫ (∇ × A)da = ∫ A ⋅ dl