Maxi Rodriguez + Torres

Originally Posted by Jeff

I agree. I think Luca Toni will be a much much better option than Torres, to be absolutely honest. Torres is not bad, he is skilful, but I won't take the gamble.

couldn't agree more.

i think milan should buy laquinta

Originally Posted by Sweden91

i think milan should buy laquinta

Man, for christ sakes. say that you're kidding, please.

no why should I?

Originally Posted by Sweden91

no why should I?

* 2 goals from open play last season, thats whats so funny

yes but his made the italian squad and he have play more then inzagi thats why it isnt funny

Originally Posted by Sweden91

yes but his made the italian squad and he have play more then inzagi thats why it isnt funny

i dont wana sound mean, but it was a miracle that he made it into the squad, most of Italy couldnt believe it, considering they left out Lucarelli... Iaquinta has speed, thats the only reason he has started more... we can do so much better than him!

Inzaghi is much more player than Iaquinta. only Lippi can't see.

yes i said we should buy him but not as a replace man for sheva i think milan need speed and i thik lippi hade his reason to take whit laquinta to wc and i think he hade preform okey i think milan could buy him cheep he have problem whit the owner in udines

yes inzagi is a mutch bether in the box than laqunta but i dont think italy need another box player they need speed

So, whats the point of having Iaquinta in Milan? we already have Inzaghi, considering that Inzaghi is the third choice of Lippi, after Iaquinta.

Oh, it's very simple, Zlatanov:

A − sin
2
A
= 2cos
2
A − 1
= 1 − 2sin
2
A
tan2A =
2 tan A
1−tan
2
A
sin3A = 3sinA − 4sin
3
A
cos3A = 4cos
3
A − 3cosA
tan3A =
3 tan A−tan
3
A
1−3 tan
2
A
sinA + sinB = 2sin
A+B
2
cos
A−B
2
4
Page 5
sinA − sinB = 2cos
A+B
2
sin
A−B
2
cosA + cosB = 2cos
A+B
2
cos
A−B
2
cosA − cosB = −2sin
A+B
2
sin
A−B
2
2sinAcosB = sin(A + B) + sin(A − B)
2cosAsinB = sin(A + B) − sin(A − B)
2cosAcosB = cos(A + B) + cos(A − B)
−2sinAsinB = cos(A + B) − cos(A − B)
asinx + b cosx = R sin(x + φ), where R = √
a
2
+ b
2
and cosφ = a/R, sinφ = b/R.
If t = tan
1
2
x then sinx =
2t
1+t
2
, cosx =
1−t
2
1+t
2
.
cosx =
1
2
(e
ix
+ e
−ix
) ;
sinx =
1
2i
(e
ix
− e
−ix
)
e
ix
= cosx + isinx ;
e
−ix
= cosx − isinx
5
Page 6
COMPLEX NUMBERS
i = √
−1
Note:- ‘j’ often used rather than ‘i’.
Exponential Notation
e
iθ
= cosθ + isinθ
De Moivre’s theorem
[r(cosθ + isinθ)]
n
= r
n
(cosnθ + isinnθ)
n
th
roots of complex numbers
If z = re
iθ
= r(cosθ + isinθ) then
z
1/n
=
n
√re
i(θ+2kπ)/n
, k = 0,±1,±2,...
HYPERBOLIC IDENTITIES
coshx = (e
x
+ e
−x
)/2
sinhx = (e
x
− e
−x
)/2
tanhx = sinhx/coshx
sechx = 1/coshx
cosechx = 1/sinhx
cothx = coshx/sinhx = 1/tanhx
coshix = cosx
sinhix = isinx
cosix = coshx
sinix = isinhx
cosh
2
A − sinh
2
A = 1
sech
2
A = 1 − tanh
2
A
cosech
2
A = coth
2
A − 1
6
Page 7
SERIES
Powers of Natural Numbers
n
k=1
k =
1
2
n(n + 1) ;
n
k=1
k
2
=
1
6
n(n + 1)(2n + 1);
n
k=1
k
3
=
1
4
n
2
(n + 1)
2
Arithmetic
S
n
=
n−1
k=0
(a + kd) =
n
2 **2a + (n − 1)d}
Geometric (convergent for −1 < r < 1)
S
n
=
n−1
k=0
ar
k
=
a(1 − r
n
)
1 − r
, S
∞
=
a
1 − r
Binomial (convergent for |x| < 1)
(1 + x)
n
= 1 + nx +
n!
(n − 2)!2!
x
2
+ ... +
n!
(n − r)!r!
x
r
+ ...
where
n!
(n − r)!r!
=
n(n − 1)(n − 2)...(n − r + 1)
r!
Maclaurin series
f(x) = f(0) + xf (0) +
x
2
2!
f (0) + ... +
x
k
k!
f
(k)
(0) + R
k+1
where R
k+1
=
x
k+1
(k + 1)!
f
(k+1)
(θx), 0 < θ < 1
Taylor series
f(a + h) = f(a) + hf (a) +
h
2
2!
f (a) + ... +
h
k
k!
f
(k)
(a) + R
k+1
where R
k+1
=
h
k+1
(k + 1)!
f
(k+1)
(a + θh) , 0 < θ < 1.
OR
f(x) = f(x
0
) + (x − x
0
)f (x
0
) +
(x − x
0
)
2
2!
f (x
0
) + ... +
(x − x
0
)
k
k!
f
(k)
(x
0
) + R
k+1
where R
k+1
=
(x − x
0
)
k+1
(k + 1)!
f
(k+1)
(x
0
+ (x − x
0
)θ), 0 < θ < 1
7
Page 8
Special Power Series
e
x
= 1 + x +
x
2
2!
+
x
3
3!
+ ... +
x
r
r!
+ ...
(all x)
sinx = x −
x
3
3!
+
x
5
5! −
x
7
7!
+ ... +
(−1)
r
x
2r+1
(2r + 1)!
+ ...
(all x)
cosx = 1 −
x
2
2!
+
x
4
4! −
x
6
6!
+ ... +
(−1)
r
x
2r
(2r)!
+ ...
(all x)
tanx = x +
x
3
3
+
2x
5
15
+
17x
7
315
+ ...
(|x| <
π
2
)
sin
−1
x = x +
1
2
x
3
3
+
1.3
2.4
x
5
5
+
1.3.5
2.4.6
x
7
7
+
... +
1.3.5....(2n − 1)
2.4.6....(2n)
x
2n+1
2n + 1
+ ...
(|x| < 1)
tan
−1
x = x −
x
3
3
+
x
5
5 −
x
7
7
+ ... + (−1)
n
x
2n+1
2n + 1
+ ... (|x| < 1)
n(1 + x) = x −
x
2
2
+
x
3
3 −
x
4
4
+ ... + (−1)
n+1
x
n
n
+ ... (−1 < x ≤ 1)
sinhx = x +
x
3
3!
+
x
5
5!
+
x
7
7!
+ ... +
x
2n+1
(2n + 1)!
+ ...
(all x)
coshx = 1 +
x
2
2!
+
x
4
4!
+
x
6
6!
+ ... +
x
2n
(2n)!
+ ...
(all x)
tanhx = x −
x
3
3
+
2x
5
15 −
17x
7
315
+ ...
(|x| <
π
2
)
sinh
−1
x = x −
1
2
x
3
3
+
1.3
2.4
x
5
5 −
1.3.5
2.4.6
x
7
7
+
... + (−1)
n
1.3.5...(2n − 1)
2.4.6...2n
x
2n+1
2n + 1
+ ...
(|x| < 1)
tanh
−1
x = x +
x
3
3
+
x
5
5
+
x
7
7
+ ...
x
2n+1
2n + 1
+ ...
(|x| < 1)
8
Page 9
DERIVATIVES
function
derivative
x
n
nx
n−1
e
x
e
x
a
x
(a > 0)
a
x
na
nx
1
x
log
a
x
1
xna
sinx
cosx
cosx
−sinx
tanx
sec
2
x
cosec x
− cosec xcotx
secx
secxtanx
cotx
− cosec
2
x
sin
−1
x
1
√1 − x
2
cos
−1
x
−
1
√1 − x
2
tan
−1
x
1
1 + x
2
sinhx
coshx
coshx
sinhx
tanhx
sech
2
x
cosech x
− cosech xcothx
sech x
− sech xtanhx
cothx
− cosech
2
x
sinh
−1
x
1
√1 + x
2
cosh
−1
x(x > 1)
1
√x
2
− 1
tanh
−1
x(|x| < 1)
1
1 − x
2
coth
−1
x(|x| > 1)
−
1
x
2
− 1
9
Page 10
Product Rule
d
dx
(u(x)v(x)) = u(x)
dv
dx
+ v(x)
du
dx
Quotient Rule
d
dx
u(x)
v(x)
=
v(x)
du
dx
− u(x)
dv
dx
[v(x)]
2
Chain Rule
d
dx
(f(g(x))) = f (g(x)) × g (x)
Leibnitz’s theorem
d
n
dx
n
(f.g) = f
(n)
.g+nf
(n−1)
.g
(1)
+
n(n − 1)
2!
f
(n−2)
.g
(2)
+...+
n!
(n − r)!r!
f
(n−r)
.g
(r)
+...+f.g
(n)
10
Page 11
INTEGRALS
function
integral
f(x)
dg(x)
dx
f(x)g(x) −
df(x)
dx
g(x)dx
x
n
(n = −1)
x
n+1
n+1
1
x
n|x|
Note:- n|x| + K = n|x/x
0
|
e
x
e
x
sinx
−cosx
cosx
sinx
tanx
n|secx|
cosec x
− n| cosec x + cotx|
or
n tan
x
2
secx
n|secx + tanx| = n tan
π
4
+
x
2
cotx
n|sinx|
1
a
2
+ x
2
1
a
tan
−1
x
a
1
a
2
− x
2
1
2a
n
a + x
a − x
or
1
a
tanh
−1
x
a
(|x| < a)
1
x
2
− a
2
1
2a
n
x − a
x + a
or
−
1
a
coth
−1
x
a
(|x| > a)
1
√a
2
− x
2
sin
−1
x
a
(a > |x|)
1
√
a
2
+ x
2
sinh
−1
x
a
or
n x +
√x
2
+ a
2
1
√x
2
− a
2
cosh
−1
x
a
or
n|x + √x
2
− a
2
|
(|x| > a)
sinhx
coshx
coshx
sinhx
tanhx
ncoshx
cosech x
− n|cosechx + cothx|
or
n tanh
x
2
sech x
2 tan
−1
e
x
cothx
n|sinhx|
11
Page 12
Double integral
f(x,y)dxdy =
g(r,s)Jdrds
where
J =
∂(x,y)
∂(r,s)
=
∂x
∂r
∂x
∂s
∂y
∂r
∂y
∂s
12
Page 13
LAPLACE TRANSFORMS
˜
f(s) =
∞
0
e
−st
f(t)dt
function
transform
1
1
s
t
n
n!
s
n+1
e
at
1
s − a
sinωt
ω
s
2
+ ω
2
cosωt
s
s
2
+ ω
2
sinhωt
ω
s
2
− ω
2
coshωt
s
s
2
− ω
2
t sinωt
2ωs
(s
2
+ ω
2
)
2
t cosωt
s
2
− ω
2
(s
2
+ ω
2
)
2
H
a
(t) = H(t − a)
e
−as
s
δ(t)
1
e
at
t
n
n!
(s − a)
n+1
e
at
sinωt
ω
(s − a)
2
+ ω
2
e
at
cosωt
s − a
(s − a)
2
+ ω
2
e
at
sinhωt
ω
(s − a)
2
− ω
2
e
at
coshωt
s − a
(s − a)
2
− ω
2
13
Page 14
Let
˜
f(s) = L{f(t)} then
L e
at
f(t)
=
˜
f(s − a),
L{tf(t)} = −
d
ds
(
˜
f(s)),
L
f(t)
t
=
∞
x=s
˜
f(x)dx if this exists.
Derivatives and integrals
Let y = y(t) and let ˜y = L{y(t)} then
L
dy
dt
= s˜y − y
0
,
L
d
2
y
dt
2
= s
2
˜y − sy
0
− y
0
,
L
t
τ=0
y(τ)dτ
=
1
s
˜y
where y
0
and y
0
are the values of y and dy/dt respectively at t = 0.
Time delay
Let
g(t) = H
a
(t)f(t − a) =
0
t < a
f(t − a) t > a
then
L{g(t)} = e
−as
˜
f(s).
Scale change
L{f(kt)} =
1
k
˜
f
s
k
.
Periodic functions
Let f(t) be of period T then
L{f(t)} =
1
1 − e
−sT
T
t=0
e
−st
f(t)dt.
14
Page 15
Convolution
Let
f(t) ∗ g(t) =
t
x=0
f(x)g(t − x)dx =
t
x=0
f(t − x)g(x)dx
then
L{f(t) ∗ g(t)} =
˜
f(s)˜g(s).
RLC circuit
For a simple RLC circuit with initial charge q
0
and initial current i
0
,
˜
E = r + Ls +
1
Cs
i − Li
0
+
1
Cs
q
0
.
Limiting values
initial value theorem
lim
t→0+
f(t) = lim
s→∞
s
˜
f(s),
final value theorem
lim
t→∞
f(t) = lim
s→0+
s
˜
f(s),
∞
0
f(t)dt = lim
s→0+
˜
f(s)
provided these limits exist.
15
Page 16
Z TRANSFORMS
Z {f(t)} =
˜
f(z) =
∞
k=0
f(kT)z
−k
function
transform
δ
t,nT
z
−n
(n ≥ 0)
e
−at
z
z − e
−aT
te
−at
Tze
−aT
(z − e
−aT
)
2
t
2
e
−at
T
2
ze
−aT
(z + e
−aT
)
(z − e
−aT
)
3
sinhat
z sinhaT
z
2
− 2z coshaT + 1
coshat
z(z − coshaT)
z
2
− 2z coshaT + 1
e
−at
sinωt
ze
−aT
sinωT
z
2
− 2ze
−aT
cosωT + e
−2aT
e
−at
cosωt
z(z − e
−aT
cosωT)
z
2
− 2ze
−aT
cosωT + e
−2aT
te
−at
sinωt
Tze
−aT
(z
2
− e
−2aT
)sinωT
(z
2
− 2ze
−aT
cosωT + e
−2aT
)
2
te
−at
cosωt
Tze
−aT
(z
2
cosωT − 2ze
−aT
+ e
−2aT
cosωT)
(z
2
− 2ze
−aT
cosωT + e
−2aT
)
2
Shift Theorem
Z {f(t + nT)} = z
n
˜
f(z) −
n−1
k=0
z
n−k
f(kT) (n > 0)
Initial value theorem
f(0) = lim
z→∞
˜
f(z)
16
Page 17
Final value theorem
f(∞) = lim
z→1
(z − 1)
˜
f(z)
provided f(∞) exists.
Inverse Formula
f(kT) =
1
2π
π
−π
e
ikθ
˜
f(e
iθ
)dθ
FOURIER SERIES AND TRANSFORMS
Fourier series
f(t) =
1
2
a
0
+
∞
n=1
{a
n
cosnωt + b
n
sinnωt}
(period T = 2π/ω)
where
a
n
=
2
T
t
0
+T
t
0
f(t)cosnωtdt
b
n
=
2
T
t
0
+T
t
0
f(t)sinnωtdt
17
Page 18
Half range Fourier series
sine series
a
n
= 0, b
n
=
4
T
T/2
0
f(t)sinnωtdt
cosine series
b
n
= 0, a
n
=
4
T
T/2
0
f(t)cos nωtdt
Finite Fourier transforms
sine transform
˜
f
s
(n) =
4
T
T/2
0
f(t)sinnωtdt
f(t) =
∞
n=1
˜
f
s
(n)sinnωt
cosine transform
˜
f
c
(n) =
4
T
T/2
0
f(t)cosnωtdt
f(t) =
1
2
˜
f
c
(0) +
∞
n=1
˜
f
c
(n)cosnωt
Fourier integral
1
2
lim
t 0
f(t) + lim
t 0
f(t) =
1
2π
∞
−∞
e
iωt
∞
−∞
f(u)e
−iωu
dudω
Fourier integral transform
˜
f(ω) = F {f(t)} =
1
√
2π
∞
−∞
e
−iωu
f(u)du
f(t) = F
−1
˜
f(ω) =
1
√
2π
∞
−∞
e
iωt
˜
f(ω)dω
18
Page 19
NUMERICAL FORMULAE
Iteration
Newton Raphson method for refining an approximate root x
0
of f(x) = 0
x
n+1
= x
n
−
f(x
n
)
f (x
n
)
Particular case to find
√
N use x
n+1
=
1
2
x
n
+
N
x
n
.
Secant Method
x
n+1
= x
n
− f(x
n
)/
f (x
n
) − f (x
n−1
)
x
n
− x
n−1
Interpolation
∆f
n
= f
n+1
− f
n
, δf
n
= f
n+
1
2
− f
n−
1
2
f
n
= f
n
− f
n−1
, µf
n
=
1
2
f
n+
1
2
+ f
n−
1
2
Gregory Newton Formula
f
p
= f
0
+ p∆f
0
+
p(p − 1)
2!
∆
2
f
0
+ ... +
p!
(p − r)!r!
∆
r
f
0
where p =
x − x
0
h
Lagrange’s Formula for n points
y =
n
i=1
y
i i
(x)
where
i
(x) =
Π
n
j=1,j=i
(x − x
j
)
Π
n
j=1,j=i
(x
i
− x
j
)
19
Page 20
Numerical differentiation
Derivatives at a tabular point
hf
0
= µδf
0
−
1
6
µδ
3
f
0
+
1
30
µδ
5
f
0
− ...
h
2
f
0
= δ
2
f
0
−
1
12
δ
4
f
0
+
1
90
δ
6
f
0
− ...
hf
0
= ∆f
0
−
1
2
∆
2
f
0
+
1
3
∆
3
f
0
−
1
4
∆
4
f
0
+
1
5
∆
5
f
0
− ...
h
2
f
0
= ∆
2
f
0
− ∆
3
f
0
+
11
12
∆
4
f
0
−
5
6
∆
5
f
0
+ ...
Numerical Integration
TrapeziumRule
x
0
+h
x
0
f(x)dx
h
2
(f
0
+ f
1
) + E
where
f
i
= f(x
0
+ ih), E = −
h
3
12
f (a), x
0
< a < x
0
+ h
Composite Trapezium Rule
x
0
+nh
x
0
f(x)dx
h
2 {f
0
+ 2f
1
+ 2f
2
+ ...2f
n−1
+ f
n
} −
h
2
12
(f
n
− f
0
) +
h
4
720
(f
n
− f
0
)...
where f
0
= f (x
0
), f
n
= f (x
0
+ nh), etc
Simpson sRule
x
0
+2h
x
0
f(x)dx
h
3
(f
0
+4f
1
+f
2
)+E
where
E = −
h
5
90
f
(4)
(a)
x
0
< a < x
0
+ 2h.
Composite Simpson’s Rule (n even)
x
0
+nh
x
0
f(x)dx
h
3
(f
0
+ 4f
1
+ 2f
2
+ 4f
3
+ 2f
4
+ ... + 2f
n−2
+ 4f
n−1
+ f
n
) + E
where
E = −
nh
5
180
f
(4)
(a).
x
0
< a < x
0
+ nh
20
Page 21
Gauss order 1. (Midpoint)
1
−1
f(x)dx = 2f (0) + E
where
E =
2
3
f (a).
− 1 < a < 1
Gauss order 2.
1
−1
f(x)dx = f −
1
√
3
+ f
1
√
3
+ E
where
E =
1
135
f
v
(a).
− 1 < a < 1
Differential Equations
To solve y = f(x,y) given initial condition y
0
at x
0
,x
n
= x
0
+ nh.
Euler’s forward method
y
n+1
= y
n
+ hf(x
n
,y
n
)
n = 0,1,2,...
Euler’s backward method
y
n+1
= y
n
+ hf(x
n+1
,y
n+1
)
n = 0,1,2,...
Heun’s method (Runge Kutta order 2)
y
n+1
= y
n
+
h
2
(f(x
n
,y
n
) + f(x
n
+ h,y
n
+ hf(x
n
,y
n
))).
Runge Kutta order 4.
y
n+1
= y
n
+
h
6
(K
1
+ 2K
2
+ 2K
3
+ K
4
)
where
K
1
= f(x
n
,y
n
)
K
2
= f x
n
+
h
2
,y
n
+
hK
1
2
K
3
= f x
n
+
h
2
,y
n
+
hK
2
2
K
4
= f(x
n
+ h,y
n
+ hK
3
)
21
Page 22
Chebyshev Polynomials
T
n
(x) = cos n(cos
−1
x)
T
o
(x) = 1
T
1
(x) = x
U
n−1
(x) =
T
n
(x)
n
=
sin[n(cos
−1
x)]
√1 − x
2
T
m
(T
n
(x)) = T
mn
(x).
T
n+1
(x) = 2xT
n
(x) − T
n−1
(x)
U
n+1
(x) = 2xU
n
(x) − U
n−1
(x)
T
n
(x)dx =
1
2
T
n+1
(x)
n + 1 −
T
n−1
(x)
n − 1
+ constant,
n ≥ 2
f(x) =
1
2
a
0
T
0
(x) + a
1
T
1
(x)...a
j
T
j
(x) + ...
where
a
j
=
2
π
π
0
f(cosθ)cos jθdθ
j ≥ 0
and f(x)dx = constant +A
1
T
1
(x) + A
2
T
2
(x) + ...A
j
T
j
(x) + ...
where A
j
= (a
j−1
− a
j+1
)/2j
j ≥ 1
22
Page 23
VECTOR FORMULAE
Scalar product a.b = abcosθ = a
1
b
1
+ a
2
b
2
+ a
3
b
3
Vector product a × b = absinθn =
i
j
k
a
1
a
2
a
3
b
1
b
2
b
3
= (a
2
b
3
− a
3
b
2
)i + (a
3
b
1
− a
1
b
3
)j + (a
1
b
2
− a
2
b
1
)k
Triple products
[a,b,c] = (a × b).c = a.(b × c) =
a
1
a
2
a
3
b
1
b
2
b
3
c
1
c
2
c
3
a × (b × c) = (a.c)b − (a.b)c
Vector Calculus
≡
∂
∂x
,
∂
∂y
,
∂
∂z
grad φ ≡ φ, div A ≡ .A, curl A ≡
× A
div grad φ ≡ .( φ) ≡
2
φ (for scalars only)
div curl A = 0
curl grad φ ≡ 0
2
A = grad div A − curl curl A
(αβ) = α β + β α
div (αA) = α div A + A.( α)
curl (αA) = α curl A − A × ( α)
div (A × B) = B. curl A − A. curl B
curl (A × B) = A div B − B div A + (B. )A − (A. )B
23
Page 24
grad (A.B) = A × curl B + B × curl A + (A. )B + (B. )A
Integral Theorems
Divergence theorem
surface
A.dS =
volume
div A dV
Stokes’ theorem
surface
( curl A).dS =
contour
A.dr
Green’s theorems
volume
(ψ
2
φ − φ
2
ψ)dV =
surface
ψ
∂φ
∂n −
φ
∂ψ
∂n |
dS|
volume
ψ
2
φ + ( φ)( ψ) dV =
surface
ψ
∂φ
∂n|
dS|
where
dS =n|dS|
Green’s theorem in the plane
(Pdx + Qdy) =
∂Q
∂x −
∂P
∂y
dxdy
24
Page 25
MECHANICS
Kinematics
Motion constant acceleration
v = u + ft,
s = ut +
1
2
ft
2
=
1
2
(u + v)t
v
2
= u
2
+ 2f.s
General solution of
d
2
x
dt
2
= −ω
2
x is
x = acosωt + bsinωt = R sin(ωt + φ)
where R = √a
2
+ b
2
and cosφ = a/R, sinφ = b/R.
In polar coordinates the velocity is (˙r,r
˙
θ) = ˙re
r
+ r
˙
θe
θ
and the acceleration is
r − r
˙
θ
2
,rθ+ 2˙r
˙
θ = (r − r
˙
θ
2
)e
r
+ (rθ+ 2˙r
˙
θ)e
θ
.
Centres of mass
The following results are for uniform bodies:
hemispherical shell, radius r
1
2
r
from centre
hemisphere, radius r
3
8
r
from centre
right circular cone, height h
3
4
h
from vertex
arc, radius r and angle 2θ
(r sinθ)/θ
from centre
sector, radius r and angle 2θ (
2
3
r sinθ)/θ from centre
Moments of inertia
i. The moment of inertia of a body of mass m about an axis = I + mh
2
, where I
is the moment of inertial about the parallel axis through the mass-centre and h
is the distance between the axes.
ii. If I
1
and I
2
are the moments of inertia of a lamina about two perpendicular
axes through a point 0 in its plane, then its moment of inertia about the axis
through 0 perpendicular to its plane is I
1
+ I
2
.
25
Page 26
iii. The following moments of inertia are for uniform bodies about the axes stated:
rod, length , through mid-point, perpendicular to rod
1
12
m
2
hoop, radius r, through centre, perpendicular to hoop
mr
2
disc, radius r, through centre, perpendicular to disc
1
2
mr
2
sphere, radius r, diameter
2
5
mr
2
Work done
W =
t
B
t
A
F.
dr
dt
dt.
26
Page 27
ALGEBRAIC STRUCTURES
A group G is a set of elements {a,b,c,...} — with a binary operation ∗ such that
i. a ∗ b is in G for all a,b in G
ii. a ∗ (b ∗ c) = (a ∗ b) ∗ c for all a,b,c in G
iii. G contains an element e, called the identity element, such that e∗a = a = a∗e
for all a in G
iv. given any a in G, there exists in G an element a
−1
, called the element inverse
to a, such that a
−1
∗ a = e = a ∗ a
−1
.
A commutative (or Abelian) group is one for which a ∗ b = b ∗ a for all a,b, in G.
A field F is a set of elements {a,b,c,...} — with two binary operations + and . such
that
i. F is a commutative group with respect to + with identity 0
ii. the non-zero elements of F form a commutative group with respect to . with
identity 1
iii. a.(b + c) = a.b + a.c for all a,b,c, in F.
A vector space V over a field F is a set of elements {a,b,c,...} — with a binary
operation + such that
i. they form a commutative group under +;
and, for all λ,µ in F and all a,b, in V ,
ii. λa is defined and is in V
iii. λ(a + b) = λa + λb
27
Page 28
iv. (λ + µ)a = λa + µa
v. (λ.µ)a = λ(µa)
vi. if 1 is an element of F such that 1.λ = λ for all λ in F, then 1a = a.
An equivalence relation R between the elements {a,b,c,...} — of a set C is a relation
such that, for all a,b,c in C
i. aRa (R is reflextive)
ii. aRb ⇒ bRa (R is symmetric)
iii. (aRb and bRc) ⇒ aRc (R is transitive).
28
Page 29
PROBABILITY DISTRIBUTIONS
Name
Parameters
Probability distribution /
Mean
Variance
density function
Binomial
n,p
P(X = r) =
n!
(n−r)!r!
p
r
(1 − p)
n−r
,
np
np(1 − p)
r = 0,1,2,...,n
Poisson
λ
P(X = n) =
e
−λ
λ
n
n!
,
λ
λ
n = 0,1,2,......
Normal
µ,σ
f(x) =
1
σ√2π
exp{−
1
2
x−µ
σ
2
},
µ
σ
2
−∞ < x < ∞
Exponential
λ
f(x) = λe
−λx
,
1
λ
1
λ
2
x > 0,
λ > 0
THE F-DISTRIBUTION
The function tabulated on the next page is the inverse cumulative distribution
function of Fisher’s F-distribution having ν
1
and ν
2
degrees of freedom. It is defined
by
P =
Γ
1
2
ν
1
+
1
2
ν
2
Γ
1
2
ν
1
Γ
1
2
ν
2
ν
1
2
ν
1
1
ν
1
2
ν
2
2
x
0
u
1
2
ν
1
−1
(ν
2
+ ν
1
u)
−
1
2
(ν
1
+ν
2
)
du.
If X has an F-distribution with ν
1
and ν
2
degrees of freedom then Pr.(X ≤ x) = P.
The table lists values of x for P = 0.95, P = 0.975 and P = 0.99, the upper number
in each set being the value for P = 0.95.
29
Page 30
ν
2
ν
1
: 1 ν
1
: 2
3
4
5
6
7
8
9
10
12
15
20
25
50 100
161 199 216 225 230 234 237 239 241 242 244 246 248 249 252 253
1
648 799 864 900 922 937 948 957 963 969 977 985 993 998 1008 1013 1
4052 5000 5403 5625 5764 5859 5928 5981 6022 6056 6106 6157 6209 6240 6303 6334
18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.46 19.48 19.49
2 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.41 39.43 39.45 39.46 39.48 39.49 2
98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.42 99.43 99.45 99.46 99.48 99.49
10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.63 8.58 8.55
3 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42 14.34 14.25 14.17 14.12 14.01 13.96 3
34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 27.05 26.87 26.69 26.58 26.35 26.24
7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.70 5.66
4 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84 8.75 8.66 8.56 8.50 8.38 8.32 4
21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.37 14.20 14.02 13.91 13.69 13.58
6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.52 4.44 4.41
5 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62 6.52 6.43 6.33 6.27 6.14 6.08
5
16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.89 9.72 9.55 9.45 9.24 9.13
5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.83 3.75 3.71
6 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 5.46 5.37 5.27 5.17 5.11 4.98 4.92
6
13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.72 7.56 7.40 7.30 7.09 6.99
5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.40 3.32 3.27
7 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 4.76 4.67 4.57 4.47 4.40 4.28 4.21
7
12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.47 6.31 6.16 6.06 5.86 5.75
5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.11 3.02 2.97
8 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30 4.20 4.10 4.00 3.94 3.81 3.74
8
11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.67 5.52 5.36 5.26 5.07 4.96
5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.89 2.80 2.76
9 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 3.96 3.87 3.77 3.67 3.60 3.47 3.40
9
10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.11 4.96 4.81 4.71 4.52 4.41
4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.73 2.64 2.59
10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.62 3.52 3.42 3.35 3.22 3.15 10
10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.71 4.56 4.41 4.31 4.12 4.01
4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.50 2.40 2.35
12 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44 3.37 3.28 3.18 3.07 3.01 2.87 2.80 12
9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.16 4.01 3.86 3.76 3.57 3.47
4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.28 2.18 2.12
15 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12 3.06 2.96 2.86 2.76 2.69 2.55 2.47 15
8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.67 3.52 3.37 3.28 3.08 2.98
4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.07 1.97 1.91
20 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84 2.77 2.68 2.57 2.46 2.40 2.25 2.17 20
8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.23 3.09 2.94 2.84 2.64 2.54
4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.09 2.01 1.96 1.84 1.78
25 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68 2.61 2.51 2.41 2.30 2.23 2.08 2.00 25
7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22 3.13 2.99 2.85 2.70 2.60 2.40 2.29
4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03 1.95 1.87 1.78 1.73 1.60 1.52
50 5.34 3.97 3.39 3.05 2.83 2.67 2.55 2.46 2.38 2.32 2.22 2.11 1.99 1.92 1.75 1.66 50
7.17 5.06 4.20 3.72 3.41 3.19 3.02 2.89 2.78 2.70 2.56 2.42 2.27 2.17 1.95 1.82
3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.97 1.93 1.85 1.77 1.68 1.62 1.48 1.39
100 5.18 3.83 3.25 2.92 2.70 2.54 2.42 2.32 2.24 2.18 2.08 1.97 1.85 1.77 1.59 1.48 100
6.90 4.82 3.98 3.51 3.21 2.99 2.82 2.69 2.59 2.50 2.37 2.22 2.07 1.97 1.74 1.60
30
Page 31
NORMAL DISTRIBUTION
The function tabulated is the cumulative distribution function of a standard N(0,1)
random variable, namely
Φ(x) =
1
√
2π
x
−∞
e
−
1
2
t
2
dt.
If X is distributed N(0,1) then Φ(x) = Pr.(X ≤ x).
x
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9773 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998
3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Iaquinta? No thanks. Inzaghi is soooo much better than him. Lippi is clueless at times....

G N L, you can fill another page with your post that nobody will be pissed at you. i promess.

Originally Posted by Sandro13Nesta

G N L, you can fill another page with your post that nobody will be pissed at you. i promess.

I was only politely giving the explanations requested by Zlatanov.