AER201 Mathematics for Aerospace Engineers Autumn Semester 2019-2020 PDF

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AER201

SCHOOL OF MATHEMATICS AND STATISTICS Mathematics for Aerospace Engineers

Autumn Semester 2019–20 2 hours

Please leave this exam paper on your desk Do not remove it from the hall

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AER201 (X, Y )

T

y pX,Y (x, y) 0 1 2 0 16/50 4/50 0 x 1 16/50 8/50 1/50 2 0 4/50 1/50 X

Y

X X

Y

I II 0.2 0.8 0.4 0.6 0.75 0.25

X

S100 =

100 X

Xi

i=1

Xi

X S100

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f (t) =



1, −1,

(0 ≤ t ≤ π ) (−π ≤ t < 0)

[−π, π] S[f ](t)

[−π, π]

∞ 4 X sin(2m − 1)t . S[f ](t) = π m=1 2m − 1

1−

(1)

1 1 1 + − + ... 3 5 7

t

g(t) = |t| (−π ≤ t ≤ π), S[g](t)

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4

Continued

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f (x, y) = x − y − 3x + 12y fx , fy , fxx, fyx , fxy .

f (x, y)



x = r cos θ, y = r sin θ. f (x, y)

 ∂f ∂f ∂f    ∂r = cos θ ∂x + sin θ ∂y ,  ∂f ∂f ∂f   . = −r sin θ + r cos θ ∂y ∂θ ∂x



∂f ∂r

2

1 + 2 r



∂f ∂θ

2



∂f = ∂x

2

+



∂f ∂y

2

.

R = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} ZZ xexy dA. R

R = {(x, y) | 0 ≤ x ≤ 1, x2 ≤ y ≤ x} ZZ 6x−2 y dA. R

R = {(x, y, z) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2x + 3y} R

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F:R →R

3

F = x y z∈R



ex+y , x2 + y, xz 2 + y 2



F =0 ∇ · (∇ × G) = 0 G : R3 → R3 f (x, y) = x2 y + xy + 1 (x, y) = (1, 0)

f (x, y) v = (2, 1)

End of Question Paper

AER201

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AER201 FORMULA SHEET Standard Probability Distributions: Name

Applications

Notation

pmf or pdf

E(X)

Var(X )

Bernoulli trial

Expt with two outcomes. Coins, constituent of more complex distributions. X ≡ no. successes

Bernoulli(p)

pX (1) = p, pX (0) = 1 − p pǫ[0, 1]

p

p(1 − p)

X ≡ no. successes in n ind. Bernoulli trials Sampling with replacement

Bin(n, p)

n! px (1 − p)n−x pX (x) = x!(n−x)! x = 0, 1, 2, ..., n pǫ[0, 1]

np

np(1 − p)

Geometric

X ≡ total no. of trials until 1st success in sequence of ind. Bernoulli trials Waiting times

Geo(p)

pX (x) = (1 − p)x−1 p x = 1, 2, ... pǫ[0, 1]

1 p

1−p p2

Poisson

Counting events occurring ‘at random’ in space or time

P o(λ)

pX (x) = e x! λ x = 0, 1, 2, ... λ>0

λ

λ

Multinomial

Generalization of Binomial to > 2 categories

multinomial(n; p1 , ..., pk )

pX (x1 , . . . , xk ) = x1 x2 n! p2 x1 !x2 !···xk ! p1

a+b 2

(b−a)2 12

Binomial

Uniform

Rounding errors Un(− 12 , 1 2)

−λ x

fX (x) =

Un(a, b)

x ∈ [a, b] a0 λ>0

Exp(λ)

Empirically, and theoretically via CLT, a good model in many situations

x

. . . pk k

1

fX (x) = n o 1 √ exp − 12 (x − µ)2 2σ 2πσ 2 x ∈ (−∞, ∞)

1

1

λ

λ2

µ

σ2

µ

Σ

fX (x1 , x2 , ..., xk ) =

1 q (2π)k |Σ |

e

−1 (X−µ) −(X−µ)T Σ 2

Bayes’ Theorem: Suppose we have two events E and F within a sample space S, then P (E|F ) =

P (F |E )P (E ) P (F )

Central Limit Theorem: Let X1 , X2 , . . . , Xn be a sequence of i.i.d random variables, each with mean µ and variance σ 2 , then for large n we have, approximately,   σ2 ¯ X(n) ∼ N µ, n or, equivalently,

  Σni=1Xi ∼ N nµ, nσ 2

Laplace transform: The Laplace transform of a function f (t) is given by: Z ∞ e−stf (t)dt. L{f (t)}(s) := 0

Properties of the Laplace transform: L{f (t)} = F (s) in the following table. L{af (t) + bg (t)} = aL{f (t)} + bL{g(t)}

linearity

L{f ′ (t)} = sF (s) − f (0)

differentiation w.r.t. t

L{f ′′(t)} = s2 F (s) − sf (0) − f ′ (0)

second differentiation w.r.t. t

L{e−kt f (t)} = F (k + s)

frequency shift

L{f (t − a)H(t − a)} = e−as F (s) L{f (at)} =

1 s) aF (a

(for a > 0)

time shift

(for a > 0)

scaling

L{f ∗ g(t)} = L{f (t)}L{g(t)} (for f (t), g(t) causal)

convolution

Table of standard Laplace transforms: f (t)

L{f (t)}(s)

Region of validity

n! sn+1

Re(s) > 0

sin(kt)

k s2 +k2

Re(s) > 0

cos(kt)

s s2 +k2

Re(s) > 0

−sT

s

Re(s) > 0

e−sT

s∈C

tn

(for n ≥ 0)

H(t − T )

(for T ≥ 0)

δ(t − T )

(for T ≥ 0)

e

Fourier transform: The Fourier transform and inverse Fourier transforms are given by: Z ∞ Z ∞ 1 f (t)e−jωt dt, f (t) = F −1 {F (ω)} = F (ω)ejωt dω. F {f (t)}(ω) = F (ω) := 2π −∞ −∞

Properties of the Fourier transform: F {f (t)} = F (ω) in the following table: F {ej θtf (t)} = F (ω − θ)

frequency shift

F {f (t − T )} = e−j ωT F (ω)

time shift

F {f (n) (t)} = (jω)n F (ω)

differentiation

F {F (t)} = 2πf (−ω)

symmetry

F {f (at)} =

ω 1 |a| F ( a )

scaling

F {f ∗ g(t)} = F {f (t)}F {g(t)}

convolution

Table of standard Fourier transforms: f (t) e−a|t|

F {f (t)}(ω)

(for a > 0)

2a a2 +ω 2

rectT (t)

sinc( T2ω )

1

2πδ(ω )

Fourier series: The Fourier series of a periodic function f (t) with fundamental period T is given by S[f ] =

∞   X a0 + an cos(ωn t) + bn sin(ωn t) 2 n=1

where 2πn , ωn = T

2 an = T

Z

T /2

−T /2

f (t) cos(ωn t)dt,

2 bn = T

Z

T /2

f (t) sin(ωn t)dt. −T /2

Coordinate systems: Cylindrical polar coordinates

Spherical polar coordinates

(x, y, z) = (r cos(θ ), r sin(θ), z) p  (r, θ, z) = x2 + y2 , arctan( yx ), z

(x, y, z) = (ρ sin(φ) cos(θ), ρ sin(φ) sin(θ), ρ cos(φ)) p  (ρ, θ, φ) = x2 + y2 + z 2 , arctan( xy ), arccos( zρ )

dV = rdrdθdz.

dV = ρ2 sin(φ)dρdφdθ....