Download ENG1091 cheat sheet completely free + more videos and notes: http://goo.gl/99vB95

1. Learnyourunicourseinoneday.Checkspoonfeedme.com forfreevideosummaries,notesandcheatsheetsbytopstudents.
CHEATSHEET
ENG1091
Mathematics for Engineering:
Algebra
Monash University, Australia
1 Vectors in 3D
Length of a vector
|v| = x2 + y2 + z2
Dot Product
v · w = vxwx + vywy + vzwz = |v||w|cosθ
Cross Product
v × w =
ˆi ˆj ˆk
vx vy vz
wx wy wz
Scalar projection v in direction of w
vw =
v · w
|w|
Vector projection v in direction of w
vw =
v · w
|w|2
w
2 Lines in 3D
Parametric Equation of a Line



x(t) = a + pt
y(t) = b + qt
z(t) = c + rt



Symmetric Form of Line Equation
x − a
p
=
y − b
q
=
z − c
r
Vector Equation of a Line
r(t) = d + tv
3 Planes in 3D
Cartesian Equation of a Plane
ax + by + cz = d
Parametric Equation of a Plane



x(u, v) = a + pu + lv
y(u, v) = b + qu + mv
z(u, v) = c + ru + nv



Vector Equation of a Plane
n · (r − d) = 0
4 Matrices
Matrix Multiplication
AB =
a b c
x y z


α ρ
β σ
γ τ

 =
aα + bβ + cγ aρ + bσ + cτ
xα + yβ + zγ xρ + yσ + zτ
Transpose


a b
c d
e f


T
=
a c e
b d f
Identity Matrix
In =







1 0 0 · · · 0
0 1 0 · · · 0
0 0 1 · · · 0
...
...
...
...
...
0 0 0 · · · 1







Symmetric and Skew Symmetric Matrix
• Symmetric: A = AT
• Skew-Symmetric: A = −AT
Basic Properties
• AB = BA
• (AB)C = A(BC)
• (AT
)T
= A
• (AB)T
= BT
AT
1

2. Learnyourunicourseinoneday.Checkspoonfeedme.com forfreevideosummaries,notesandcheatsheetsbytopstudents.
CHEATSHEET
5 Determinants
2x2 Matrix
det
a b
c d
=
a b
c d
= ad − cb
3x3 Matrix
a b c
d e f
g h i
= a
e f
h i
− b
d f
g i
+ c
d e
g h
6 Inverse Matrices
2x2 Matrix
A−1
=
a b
c d
−1
=
1
ad − bc
d −b
−c a
Inverse Using Gaussian Elimination
Step 1 Augment with identity matrix [A|I]
Step 2 Row reduce to get [I|A−
1]
Inverse Using Determinants
Step 1 Select a row I and column J of matrix A
Step 2 Compute (−1)i+j (SJI)
det(A)
Step 3 Store at row J column I in inverse matrix
Step 4 Repeat for all entries
7 Eigenvalues and Eigenvectors
Av = λv
A is a square matrix, v is the eigenvector (a vector), λ is
the eigenvalue (a scalar).
Characteristic Equation
0 = det(A − λI) =
a1,1 − λ a1,2 · · · a1,n
a2,1 a2,2 − λ · · · a2,n
a3,1 a3,2 · · · a3,n
...
...
...
...
an,1 an,2 · · · an,n − λ
Eigendecomposition
Given matrix A with eigenvectors u, v, w and eigenvalues
α,β,γ:
A =


u1 v1 w1
u2 v2 w2
u3 v3 w3




α 0 0
0 β 0
0 0γ




u1 u2 u3
v1 v2 v3
w1 w2 w3


Powers/Inverses Using Eigendecomposition
An
=


u1 v1 w1
u2 v2 w2
u3 v3 w3




αn
0 0
0 βn
0
0 0γn




u1 u2 u3
v1 v2 v3
w1 w2 w3


8 Hyperbolic Functions
sinh x =
eu
− e−u
2
cosh x =
eu
+ e−u
2
Properties of Hyperbolic Functions
cosh2
x − sinh2
x = 1
d
dx
(cosh x) = sinh x
d
dx
(sinh x) = cosh x
cosh(u + v) = cosh u cosh v + sinh u sinh v
sinh(u − v) = sinh u cosh v − sinh v cosh u
2 cosh2
x = 1 + cosh 2x
2 sinh2
x = −1 + cosh 2x
9 Integration
Basic Integrals to Remember
ex
dx = ex
+ C
cos xdx = sin x + C
sin xdx = − cos x + C
xn
dx =
xn+1
n + 1
+ C (n = −1)
1
x
dx = loge(x) + C
9.1 Integration By Substitution
I = f(x) dx = f(x(u))
dx
du
du
2

3. Learnyourunicourseinoneday.Checkspoonfeedme.com forfreevideosummaries,notesandcheatsheetsbytopstudents.
CHEATSHEET
Integration By Parts
u dv = uv − v du
Improper Integrals
Examples:
∞
0
x dx or
1
0
1
x
dx or
1
−1
1
x
dx
Strategy:
lim
→∞ 0
x dx or lim
→0
1
1
x
dx
10 Series and Sequences
Arithmetic
an = an−1 + d = a + nd
Sn = (n + 1)a +
n(n + 1)d
2
Fibonacci
Fn = Fn − 1 + Fn − 2
Geometric
an = san−1 = a0sn
Partial sums for geometric series
Sn =



(n + 1)a0 : s = 1
(1 − sn+1
)a0
1 − s
: s = 1
Power Series
In x around the point x = a:
a0 + a1(x − a) + a2(x − a)2
+ ... + an(x − a)n
+ ...
= Σ∞
n=0an(x − a)n
Maclaurin Series
f(x) = a0 + a1x + a2x2
+ ... + anxn
+ ...
with:
an =
1
n!
dn
f
dxn
x=0
Taylor Series
If f(x) is infinitely differentiable at x=0, then:
f(x) = a0 + a1(x − 1) + a2(x − a)2
+ ... + an(x − a)n
+ ...
with:
an =
1
n!
dn
f
dxn
x=a
11 Convergence Tests
For Improper Integrals
Given I =
∞
0
f(x) dx f(x) > 0
Convergence
Find c(x) such that
1. 0 < f(x) < c(x)
2. lim
→∞ 0
c(x) dx is finite
Divergence
Find d(x) such that
1. 0 < d(x) < f(x)
2. lim
→∞ 0
c(x) dx = ∞
Comparison Test
If C = Σ∞
n=0cn is a convergent series and D = Σ∞
n=0dn a
divergent one then:
Convergence: When 0 < an < cn ∀n > m
Divergence: When 0 < dn < an ∀n > m
Integral Test
Given S = Σ∞
k=0ak with ak = f(k) take:
lim
n→∞
<
∞
0
f(x) dx < lim
n to∞
Sn−1
Convergence: Improper integral converges
Divergence: Improper integral diverges
Ratio Test
Take:
L = lim
n→∞
an+1
an
Then:
Convergence: L < 1
Divergence: L > 1
Indeterminate: L = 1
3

4. Learnyourunicourseinoneday.Checkspoonfeedme.com forfreevideosummaries,notesandcheatsheetsbytopstudents.
CHEATSHEET
Alternating Series
If |an| → ∞:
Convergence: When |an+1| < |an| ∀n > m
Divergence: Otherwise
Absolute Convergence
If Σ∞
n=0|an| converges then Σ∞
n=0an converges absolutely,
otherwise we can’t say anything.
12 Taylor Series
Useful Functions
ex
= 1 + x +
x2
2!
+
x3
3!
+ ... = Σ∞
n=0
xn
n!
cos x = 1 −
x2
2!
+
x4
4!
−
x6
6!
+ ... = Σ∞
n=0(−1)n x2n
2n!
sin x = x −
x3
3!
+
x5
5!
−
x7
7!
+ ... = Σ∞
n=0(−1)n+1 x2n+1
(2n + 1)!
Error Bound
En = |f(x) − Pn(x)| <
M
(n + 1)
Rn+1
M is the maximum value of |f(n+1)
(x)| in |x − a| < R
With Remainder Term
f(x) = Pn(x) + En(x)
With:
Pn(x) = f(a) + (x − a)f (a) +
(x − a)2
2!
f (a) + ...
+
(x − a)n
n!
f(n)
(a)
And
En(x) =
x
a
(−1)n (u − x)n
n!
f(n+1)
(u) du
13 l’Hopital’s Rule
lim
x→a
f(x)
g(x)
= lim
x→a
f (x)
g (x)
lim
x→a
f(x) = lim
x→a
g(x) = 0 or = ∞
14 1st Linear Order ODEs
Separable
If
dy
dx
=
f(x)
g(y)
, set
g(y) dy = f(x) dx
Integrating Factor
To solve:
dy
dx
+ P(x)y = Q(x)
Use:
y(x) =
1
I(x)
I(x)Q(x) dx
Where:
I(x) = e P (x) dx
15 2nd Order Linear ODEs
Linear, Homogeneous
For:
a
d2
y
dx2
+ b
dy
dx
+ cy = 0
Characteristic Equation:
aλ2
+ bλ + c = 0
If λ1 and λ2 are the roots:
Case 1: λ1 = λ2 and real ⇒ y(x) = Aeλ1x
+ Beλ2x
Case 2: λ = α ± iβ ⇒ y(x) = eαx
(A cos(βx) +
B sin(βx))
Case 3: λ1 = λ2 ⇒ y(x) = (A + Bx)eλx
4

5. Learnyourunicourseinoneday.Checkspoonfeedme.com forfreevideosummaries,notesandcheatsheetsbytopstudents.
CHEATSHEET
Non-homogeneous
Method of Undetermined Coefficients Given:
a
d2
y
dx2
+ b
dy
dx
+ cy = S(x)
Guesses for various S(x):
For:
S(x) = (a + bx + cx2
+ ... + dxn
)ekx
Try:
yp(x) = (e + fx + gx2
+ ... + hxn
)ekx
For:
S(x) = (a sin(bx) + c cos(bx))ekx
Try:
yp(x) = (d cos(bx) + f sin(bx))ekx
16 Coupled Systems of ODEs
Given the homogeneous equation ˙y = Ay:
Step 1: Find eigenvalues of A
Step 2: For each eigenvalue λ find the eigenvector v
Step 3: For each pair (λ, v) a solution is given by the
Euler’s Guess y = eλx
v
17 Functions of Several Variables
Chain Rule
Along a path x = x(s), y = y(s):
df
ds
=
∂f
∂x
dx
ds
+
∂f
∂y
dy
ds
If x = x(u, v) and y = y(u, v) then:
∂f
∂u
=
∂f
∂x
∂x
∂u
+
∂f
∂y
∂y
∂u
∂f
∂v
=
∂f
∂x
∂x
∂v
+
∂f
∂y
∂y
∂v
Directional Derivative
Of function f in direction of unit vector t:
df
ds
= t · f = tf
where:
f =
∂f
∂x
ˆi +
∂f
∂y
ˆj
Tangent Plane
The tangent plane to f = f(x, y) at P is:
˜f(x, y) = fP + (x − a)
∂f
∂x P
+ (y − b)
∂f
∂y P
Finding and Classifying Extrema
Extrema at:
f = 0
Let:
D =
∂2
f
∂x2
∂2
f
∂y2
−
∂2
f
∂x∂y
2
Then:
Local Minima: D ≥ 0 and
∂2
f
∂x2
> 0
Local Maxima: D ≥ 0 and
∂2
f
∂x2
< 0
Saddle Point: D < 0
5

6. Learnyourunicourseinoneday.Checkspoonfeedme.com forfreevideosummaries,notesandcheatsheetsbytopstudents.
CHEATSHEET