CHEATSHEET
MATH1110
Scalar projection v in direction of w
Mathematics 1
vw =
University of Newcastle
1
Properties of Scalar Triple
v·w
|w|
[u, v, w] = −[u, w, v] = −[v, u, w]
[u, v, w] = [v, w, u] = [w, u, v]
Vector projection v in direction of w
Vectors
vw =
Length of a vector
|v| =
p
x2 + y 2 + z 2
v·w
|w|2
[λu, v, w] = λ[u, v, w]
w
2
Triple Products


x(t) = a + pt
y(t) = b + qt


z(t) = c + rt
u × (v × w)
v · w = vx wx + vy wy + vz wz = |v||w|cosθ
Scalar Triple
Basic Properties of Dot Product
[u, v, w] = u · (v × w)
v · v = ||v||2
v×w
v·w =w·v
u · (v + w) = u · v + u · w
y−b
z−c
x−a
=
=
p
q
r
Vector Equation of a Line
w
r(t) = d + tv
v
Cross Product
î
v × w = vx
wx
Symmetric Form of Line Equation
V = [u, v, w]
u
(λv) · w = λ(v · w) = v · (λw)
Lines in 3D
Parametric Equation of a Line
Vector Triple
Scalar (Dot) Product
[(a + b), v, w] = [a, v, w] + [b, v, w]
ĵ
vy
wy
k̂
vz
wz
Volume of a parallelpiped
3
Properties of Triple Products
Cartesian Equation of a Plane
(u · v)w 6= u(v · w)
Basic Properties of Cross Product
Planes in 3D
ax + by + cz = d
u · (v × w) = (u × v) · w
u × v = −v × u
u · (v × w) = v · (w × u) = w · (u × v)
u × (v + w) = u × v + u × w
u × (v × w) 6= (u × v) × w
λ(v × w) = (λv) × w = v × (λw)
u × (v × w) = (u · w)v − (u · v)w
Parametric Equation of a Plane


 x(u, v) = a + pu + lv 
y(u, v) = b + qu + mv


z(u, v) = c + ru + nv
1
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CHEATSHEET
Vector Equation of a Plane
n · (r − d) = 0
Distance of Point From Plane
d=
|n · P0 P1 |
||n||
Angle Between Planes
cos θ =
4
n1 · n2
||n1 ||||n2 ||
Complex Numbers
z = a + ib
Where:
i2 = −1
Properties of Modulus and Conjugate
|zw| = |z||w|
5
Trigonometry
Angle Sum Formulae
z |z|
w |w|
sin(θ + φ) = sin θ cos φ + sin φ cos θ
cos(θ + φ) = cos θ cos φ − sin θ sin φ
|z + w| ≤ |z| + |w|
tan(θ + φ) =
|z − w| ≥ |z| − |w|
tan θ + tan φ
1 − tan θ tan φ
z±w =z±w
Pythagorean Identity
zw = zw
cos2 x + sin2 x = 1
Polar Form
z = rcisθ = r(cos θ + i sin θ)
Where r is the modulus ||z|| and θ the argument arg z .
6
Hyperbolic
tions
Trigonometric
Func-
Polar Form Operations and Properties
rcis(θ) × pcis(φ) = rpcis(θ + φ)
Modulus
|z| =
p
a2 + b2
y=0.5
r
rcis(θ)
= cis(θ − φ)
pcis(φ)
p
arg(zw) = arg(z) + arg(w)
z
= arg(z) − arg(w)
arg
w
Conjugate
If
z = a + ib
then
Euler’s Formula
cosh(x) =
ex + e−x
2
sinh(x) =
ex − e−x
2
eix = cos x + i sin x
z = z − ib
Where e is Euler’s Number and x is real.
De Moivre’s Theorem
Given z = reiθ :
n
n inθ
z =r e
tanh x =
sinh x)
cosh x
2
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CHEATSHEET
Properties of Hyperbolic Functions
2
9 Inverse Trigonometric Functions
2
cosh x − sinh x = 1
cosh(x + y) = cosh x cosh y + sinh x sinh y
π
2π
π
2
3π
2
sinh(x − y) = sinh x cosh y − sinh y cosh x
tanh x + tanh y
1 + tanh x tanh y
−0.5
0.5
−
Function Composition
Power Function
π
2
π
2
−1
−0.5
y = arcsin x
0.5
y = arccos x
(g ◦ f )(x) = g(f (x))
1
d[xn ]
= nxn−1
dx
Addition Rule
d[u(x) + v(x)]
= u0 (x) + v 0 (x)
dx
π
2
8
d[kf (x)]
= kf 0 (x)
dx
1
−π
7
Scalar Multiple
π
−1
tanh(x + y) =
Rules for Differentiation
Logarithms
Product Rule
−3
−2
−1
Basic Properties
1
−
2
3
π
2
loga 1 = 0
Quotient Rule
loga a = 1
loga xy = loga x + loga y
loga
x
= loga x − loga y
y
loga xn = n loga x
y = arctan x
If f (x) =
10
Differentiation
ax = bx logb a
loga x =
logb x
logb a
u(x)
then:
v(x)
f 0 (x) =
Average Rate of Change
Of f (x) over the interval [x1 , x2 ]:
Change of Base
d[u(x)v(x)]
= u(x)v 0 (x) + u0 (x)v(x)
dx
f (x2 ) − f (x1 )
Avg. rate of change =
x2 − x1
v(x)u0 (x) − u(x)v 0 (x)
[v(x)]2
Chain Rule
If f (x) = u[v(x)] then:
f 0 (x) = v 0 (x)u0 (x)
First Principles
That is:
Derivative of f (x) with respect to x at x = a:
df du
df
=
dx
du dx
dy
f (a + h) − f (a)
= lim
dx h→a
h
3
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CHEATSHEET
Exponential Function
x
d[e ]
= ex
dx
Integration
Stationary Points
Lower Riemann Sum Formula
Over interval [a, b] with n rectangles:
Concavity
1
d[ln x]
= for x > 0
dx
x
A=
• f 00 (x) > 0: Concave up - local minimum
• f 00 (x) < 0: Concave down - local maximum
Trig Functions
d[cos x]
= − sin x
dx
Point of Inflection
n
X
f (xk )∆x
k=1
Where: ∆x =
b−a
n
Integral as Riemann Summ
f 00 (x) = 0
b
Z
f (x) dx = lim
11
Hyperbolic Functions
d[sinh x]
= cosh x
dx
1
d[tanh x]
=
dx
cosh2 x
12
f 0 (x) = 0
Logarithmic Functions
d[sin x]
= cos x
dx
d[tan x]
= sec2 x
dx
Derivative Tests
d[cosh x]
= − sinh x
dx
Inverse Trig Functions
d[arcsin x]
1
=√
for − 1 < x < 1
dx
1 − x2
d[arccos x]
−1
=√
for − 1 < x < 1
dx
1 − x2
1
d[arctan x]
=
dx
1 + x2
Linear Approximations
Linearisation at a Point
Linearisation of f (x) at x = a is given:
0
L(x) = f (a)(x − a) + f (a)
n→∞
a
Where: ∆x =
n
X
f (xk )∆x
k=1
b−a
n
Fundamental Theorem of Calculus
Given f is continuous over [a, b]
Z
x
f (u) du then F 0 (x) = f (x)
Small Change Formula
1. If F (x) =
Change in a function value ∆f when perturbed by a small
amount ∆x is:
2. If G(x) is any antiderivative of f (x) then:
a
∆f ≈ f 0 (a)∆x
Z
b
f (x) dx = G(b) − G(a)
a
Integration By Substitution
Z
I=
Z
f (x) dx =
f (x(u))
dx
du
du
4
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CHEATSHEET
Integration By Parts
Z
Z
u dv = uv −
v du
Standard Integrals
Z
xn+1
+ C (n 6= −1)
xn dx =
n+1
Z
cos x dx = sin x + C
Z
sin x dx = − cos x + C
Z
2
sec x dx = − tan x + C
Z
x
x
e dx = e + C
Z
1
dx = loge (x) + C
x
Z
cosh x dx = sinh x + C
Trapezoidal Rule
14
∆x
{f (x0 ) + 2f (x1 ) + ... + 2f (xn−1 + f (xn )}
2
b−a
Where: ∆x =
n
Force
Simpson’s Rule
Where:
Tn =
b−a
Where: ∆x =
n
13
Volumes
Z
1
dx = arctan x + C
1 + x2
A is area of object
d is depth
F
= ρgd
A
Of Revolution
x-Axis Rotation
b
Z
π(f (x))2 dx
a
sinh x dx = cosh x + C
1
dx = arcsin x + C
1 − x2
g is gravitational acceleration (≈ 9.8)
P =
V =
√
ρ is density
Pressure
Z
Z
F = ρgAd
∆x
{f (x0 ) + 4f (x1 ) + 2f (x2 ) + ... + 4f (xn−1 ) + f (xn )}
3
∆x
=
{f (x0 ) + 4[Sum of odds] + 2[Sum of evens] + ...
3
... + f (xn )}
Sn =
Applications: Hydrostatic Pressure and Force
y-Axis Rotation
b
Z
V =
2πxf (x) dx
a
By Slices
Z
V =
b
A(x) dx
a
Where: A(x) is the cross-sectional area.
5
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