36 Aeronautical Engineer’s Data Book Square matrix This is a matrix having the same number of rows and columns.. Diagonal matrix This is a square matrix in which all the elements are z
Trang 1�+
3
33 33 33
� �
31 Fundamental dimensions and units
33 (A 2 B) = A 2 33 + 33 2 B
= – 33 (B 2 A)
dt Gradient
The gradient (grad) of a scalar field +(x, y, z) is
∂
∂
∂
∂
i 33 + j 33 + k 3∂
� ∂
grad + = 6+ =
∂
j
3 3
∂
i +
∂
∂ 3
=
= V x (x, y, z) i + V y (x, y, z) j + Vz
∂ +
3 3
∂ +
3 3
∂
div = 6· 3Vx Vy V
∂x ∂y ∂z
Curl
Curl (rotation) is:
z
curl V = 6 2 V = ∂ ∂ ∂
Vx Vy Vz
∂
3 3–
∂
3 Vz V
∂y ∂z
∂
3 3–
∂
3 Vx V
∂z ∂x
=
∂
3 3–
∂
3 Vy V
∂x ∂y k
x +
2.8.8 Differentiation
Rules for differentiation: y, u and v are
functions of x; a, b, c and n are constants
33 (au ± bv) = a 33 ± b 33
33 = u 33 + v 33
33 33 = 33 33 – 33 33
Trang 23 3
3
3
3
d
3
x
3
d (u
n ) = nu n–1 u
d3
x
d
x
du
3
d
3
u
n
3
x
3
u
= –
n
= 1/ ,
x
3
3
d
3
x
d
3 if 3dx3
d ≠ 0
u
d
3
x
3
d
u
f (u) = f’(u)
d3
x
d
3
�x
d
3
x
3
d a f(t)dt = f(x)
�b
d
3
x
3
d x f(t)dt = – f(x)
�b
f(x, t)dt = �b
f
3
∂
∂x
d
3
x
3
�v
f(x, t)dt = �u
3
x
d
3
f
3 dt + f (x, v)
d
3
x
3
u
3
x
d
3
– f (x, u)
d Higher derivatives
d
3 2 y
dx2
d
3
x
d
3
� y
Second derivatives = 3
x
3
d
= f"(x) = y"
3 2
dx2
d
3
x
d
3
�d �2
+ f '(u)
3
d 3 f(u) = f "(u)2
x
Derivatives of exponentials and logarithms
d
(ax + b) n = na(ax + b) n–1
3
x
3
d
d
3
x
3
d e
ax = ae ax
d
3
x
3
d ln ax = 3
x
1
3 , ax > 0
Trang 33
33 Fundamental dimensions and units
d
3
x
3
d a
u = a u ln a u
d3
x
d
3
d3
x
d
3 3
x
3
d loga u = log a e
1 3
u
Derivatives of trigonometric functions in radians
sin x = cos x, cos x = – sin x
3
x
3
x
3
d
d
3
x
3
d tan x = sec
2 x = 1 + tan2 x
d
3
x
3
d cot x = –cosec
2x
3
x
3
x
3
c
sec x = 2
os = sec x tan x
3
x
3
x
3
s
cosec x = –
in2 = – cosec x cot x
arcsin x = –
3
x
3
x
3
d arccos x
1
= 33 for angles in the(1 – x2)1/2 first quadrant
Derivatives of hyperbolic functions
sinh x = cosh x, cosh x = sinh x
3
x
3
x
3
d
tanh x = sech2 cosh x = – cosech2 3
x
3
x
3
d
(arcsinh x) = 32 +1)1/2
3
x
3
(arccosh x) = 3
1)1/2
3
x
3
x
Trang 43
x
Partial derivatives Let f(x, y) be a function of
the two variables x and y The partial deriva tive of f with respect to x, keeping y constant is:
f (x + h, y) – f (x, y)
∂
Similarly the partial derivative of f with respect
to y, keeping x constant, is
f
3
∂ 3 3
v
∂
3
∂y k→0
Chain rule for partial derivatives To change
variables from (x, y) to (u, v) where u = u(x, y),
v = v(x, y), both x = x(u, v) and y(u, v) exist and f(x, y) = f [x(u, v), y(u, v)] = F(u, v)
f
3
3
u
3
∂
3 = 3
f (x, y + k) – f (x, y)
f
3
∂
3
f
3 y
∂v ∂v ∂v ∂y
f
3
y
3
f
x
∂u ∂x ∂u ∂y
f
3
∂
∂x 3x
∂
3 u
∂ 3 3
x
∂ 3
∂
3 ∂ F + ∂ ∂ , v F ∂y ∂ 3f 3y
∂
3 u
∂ 3 3
y
∂ 3
∂
3 F + v F
∂ ∂ ∂
2.8.9 Integration
a+1
x
a ≠ –1
1
3
a
e
3
+
kx
k
,
kx
e
a x
a > 0, a ≠ 1
a
3
ln ,
ln x x ln x – x
tan x ln | sec x |
cot x ln | sin x |
sec x ln | sec x + tan x |
3 1 3
tan 3 (x + 3 π) |
Trang 533
33
2
35 Fundamental dimensions and units
3 2 1
3 x |
ln | tan
cosec x
sin2x 323(x – sin 2x)323
2 3213(x + sin 2x)3213
cos x
sec2 x tan x
sinh x cosh x
cosh x sinh x
tanh x ln cosh x
sech x 2 arctan e x
3 2 1 3
cosech x ln | tanh x |
sech2 x tanh x
3a2+ x
1
a
1
a
x
3 , a ≠ 0
arctan
1 a – x
�– 3 ln a ≠ a
a
3
x
3
a + ,
x – a
32
3
a
1
a ≠ 0
a
3
x +
3
a
3
a ≠ 0 arcsin
2)1/2
|
3
| a ,
�ln [x + (x2 – a2)1/2]
1
2)1/2
a
x
3 , a ≠ 0
arccosh
2.8.10 Matrices
A matrix which has an array of m 2 n numbers arranged in m rows and n columns is called an
m 2 n matrix It is denoted by:
� a11 a12 a 1n
a
a21 a22
�
a m1 a m2 a mn
Trang 636 Aeronautical Engineer’s Data Book
Square matrix
This is a matrix having the same number of rows and columns
a11 a12 a13
a21 a22 a23 is a square matrix of order 3 2
a31 a32 a33 3
Diagonal matrix
This is a square matrix in which all the elements are zero except those in the leading diagonal
�a11 0 0
0�
0 a22 is a diagonal matrix of order 3
0 0 a33 2 3
Unit matrix
This is a diagonal matrix with the elements in the leading diagonal all equal to 1 All other elements are 0 The unit matrix is denoted
by I
1 0 0
0 1 0
I = � �0 0 1
Addition of matrices
Two matrices may be added provided that they are of the same order This is done by adding the corresponding elements in each matrix
a11 a12 a13�+ b11 b12 b
�a21 a22 a23 �b21 b22 b23
a11 + b11 a12 + b12 a13 + b
= �a21 + b21 a22 + b22 a23 + b23
Subtraction of matrices
Subtraction is done in a similar way to addition except that the corresponding elements are subtracted
a11 a12 b11 b12 a11 – b11 a12 –b12
� �a21 a22 – � �b21 b22 = �a21 –b21 a22 –b22�
Trang 7� � � �
37 Fundamental dimensions and units
Scalar multiplication
A matrix may be multiplied by a number as follows:
a11 a12 ba11 ba12
b� �a21 a22 = �ba21 ba22�
General matrix multiplication
Two matrices can be multiplied together provided the number of columns in the first matrix is equal to the number of rows in the second matrix
b11 b12
a11 a12 a13
b21 b22
a21 a22a23
b31 b32
a11b11+a12b22 +a13b31 a11b12 +a12b22 +a13b32
= �a21b11 +a22b21 +a23b31 a21b12 +a22b22 +a23b32�
If matrix A is of order (p 2 q) and matrix B is
of order (q 2 r) then if C = AB, the order of C
is (p 2 r)
Transposition of a matrix
When the rows of a matrix are interchanged
with its columns the matrix is said to be trans
posed If the original matrix is denoted by A, its
transpose is denoted by A' or A T
a11 a12 a13
then A T = � �
a
21 a22a
a13 a23 Adjoint of a matrix
If A =[a ij ] is any matrix and A ij is the cofactor
of a ij the matrix [A ij]T is called the adjoint of A
Thus:
a
a
21 a22 a 2n A12 A22 A n2
11 a12 a 1n
adj A = �A11 A21 A n1
A = �
�
a n1 a n2 a mn A1n A2n A nn
Trang 8�
Singular matrix
A square matrix is singular if the determinant
of its coefficients is zero
If A is a non-singular matrix of order (n then its inverse is denoted by A–1 such that AA
= I = A–1 A
adj (A)
A–1 = 33 ∆ = det (A)∆
A ij = cofactor of a ij
a
a11 a12 a 1n
� �A11 A21 A n1
21 a22 a 2n A12 A22 A n2
A–1 = 31 3 . .
If A = � ∆
a
n1 a n2 a nn A 1n A 2n A nn
2.8.11 Solutions of simultaneous linear equations
The set of linear equations
a11x1+ a12x2 + + a 1n x n = b1
a21x1+ a22x2 + + a 2n x n = b2
a n1 x1+ a n2 x2 + + a nn x n = b n
a
where the as and bs are known, may be repre sented by the single matrix equation Ax = b, where A is the (n 2 n) matrix of coefficients,
ij , and x and b are (n 2 1) column vectors
The solution to this matrix equation, if A is non-singular, may be written as x = A–1b
which leads to a solution given by Cramer’s
rule:
x i = det D i /det A i = 1, 2, , n
where det D i is the determinant obtained from
det A by replacing the elements of a ki of the ith column by the elements b k (k = 1, 2, , n) Note that this rule is obtained by using A–1 = (det A)–1
adj A and so again is of practical use only when
n ≤ 4
Trang 9+ +
x
39 Fundamental dimensions and units
If det A = 0 but det D i ≠ 0 for some i then the equations are inconsistent: for example, x + y =
2, x + y = 3 has no solution
2.8.12 Ordinary differential equations
A differential equation is a relation between a
function and its derivatives The order of the
highest derivative appearing is the order of the
differential equation Equations involving
only one independent variable are ordinary
differential equations, whereas those involv
ing more than one are partial differential
equations
If the equation involves no products of the function with its derivatives or itself nor of
derivatives with each other, then it is linear Otherwise it is non-linear
A linear differential equation of order n has
the form:
d n–1
3
x
d
3 3
1
3
d
where P i (i = 0, 1 ., n) F may be functions of
x or constants, and P0 ≠ 0
First order differential equations
3 n
d
3 n
P0 y
dx + P1 + + P n–1 + P n y = F
3
y
d
3
d = f 3
x
y
x
y
3 homo- substitute u =
dy
3 )
3
g
geneous
∫
(y
3
x
d
3y = f(x)g(y) separable
note that roots of
g(y) = 0 are also
solutions
∂
3∂ 3∂ ∂
y
3
x
d
3
+ f(x, y) = 0 exact = g
and solve these
3
x
∂
3g
= ∂
equations for +
+ (x, y) = constant
is the solution
f
3
∂
y
and3
Trang 1040 Aeronautical Engineer’s Data Book
dy
3
dx 3 + f(x)y linear Multiply through by
p(x) = exp(∫x f(t)dt)
p(x)y = ∫x g(s)p(s)ds
+ C
Second order (linear) equations
These are of the form:
P0(x) 33 + P1 (x) 33 + P2 (x)y = F(x)
When P0, P1, P2 are constants and f(x) = 0, the
solution is found from the roots of the auxiliary equation:
P0m2+ P1m + P2= 0
There are three other cases:
(i) Roots m = and are real and ≠
y(x) = Ae x + Be x
(ii) Double roots: =
x
y(x) = (A + Bx)e
(iii) Roots are complex: m = k ± il
y(x) = (A cos lx + B sin lx)e kx
2.8.13 Laplace transforms
If f(t) is defined for all t in 0 ≤ t < ∞, then
L[f(t)] = F(s) = �∞
e –st f(t)dt
0
is called the Laplace transform of f(t) The two functions of f(t), F(s) are known as a transform
pair, and
f(t) = L–1[F(s)]
is called the inverse transform of F(s)
c1f(t) + c2g(t) c1F(s) + c2G(s)
Trang 1133
33
33
33
�
�t
0
(–t) n f(t)
n
s n
d
F
3
d
3
e at f(t)
e
F(s – a) –as F(s) f(t – a)H(t – a)
n
t n
d
f
3
d
3
3
a
1
3 e –bt sin at, a > 0
n
s n F(s) – � s n–r f (r–1) (0+)
r=1
1 2
(s = b)2 + a
s + b –bt
(s + b )2 + a
1
3
a
1
3 e –bt sinh at, a > 0
(s + b)2 + a2
s + b
2
(πt)–1/2
n t n–1/2
s
(s + b)2 + a
–1/2
s –(n+1/2)
33,
1·3·5 (2n –1)�π
2
p(– t)
1/2
2(π 3)
4
ex a
(a > 0) e –a� s
t
2.8.14 Basic trigonometry
Definitions (see Figure 2.9)
sine: sin A = 3
r
y
3 cosine: cos A = 3
r
x
3
3
x
y
3 cotangent: cot A = 3
y
x
3
tangent: tan A =
r 33
y
secant: sec A = cosecant: cosec A =
Trang 127 3
A
y
r
x
Fig 2.9 Basic trigonometry
Relations between trigonometric functions
sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A
sin A = s cos A = c tan A = t
sin A s (1 – c2)1/2 t(1 + t2)–1/2
cos A (1 – s2)1/2 c (1 + t2)–1/2
tan A s(1 – s2)1/2 (1 – c2)1/2/c t
A is assumed to be in the first quadrant; signs
of square roots must be chosen appropriately in other quadrants
Addition formulae
sin(A ± B) = sin A cos B ± cos A sin B cos(A ± B) = cos A cos B 7 sin A sin B
tan A ± tan B tan(A ± B) = 3
tan A tan B
Sum and difference formulae
3 2 1
sin A + sin B = 2 sin (A + B) cos (A – B)
3 2 1
sin A – sin B = 2 cos (A + B) sin (A – B)
3 2 1
cos A + cos B = 2 cos (A + B) cos (A – B)
3 1
cos A – cos B = 2 sin (A + B) sin (B – A)
Trang 1343 Fundamental dimensions and units
Product formulae
3 2 3
sin A sin B = {cos(A – B) – cos(A + B)}
3 2 1 3
cos A cos B = {cos(A – B) + cos(A + B)} sin A cos B =323{sin(A – B) + sin(A + B)}
Powers of trigonometric functions
sin2 A =3213– 3213cos 2A
2 A =3213 3213 cos 2A
cos
sin3 A =3433sin A – sin 3A 3413
3 A =3433cos A +3413cos 3A
cos
2.8.15 Co-ordinate geometry
Straight-line
General equation
ax + by + c = 0
m = gradient
c = intercept on the y-axis
Gradient equation
y = mx + c
Intercept equation
x
3
A 3y3
B
A = intercept on the x-axis
= 1
B = intercept on the y-axis
Perpendicular equation
x cos + y sin = p
p = length of perpendicular from the origin to the line
= angle that the perpendicular makes
with the x-axis
The distance between two points P(x1, y1) and
Q(x2, y2) and is given by:
�2)2 + (�)2
PQ = �(x1– x�y1– y2�
The equation of the line joining two points (x1,
y1) and (x2, y2) is given by:
3
3
y
y – y1
– y
x – x
3
3
x
1
=
– x
Trang 14�
Circle
General equation x2– y2+ 2gx + 2fy + c = 0 The centre has co-ordinates (–g, –f)
The radius is r = �g�2+ f 2–c
The equation of the tangent at (x1, y1)
to the circle is:
xx1+ yy1+ g(x + x 1 ) + f(y + y 1 ) + c = 0
The length of the tangent from to the circle is:
t2= x1+ y1+ 2gx1+ 2fy1+ c
Parabola (see Figure 2.10)
SP
Eccentricity = e = 33 = 1
PD
With focus S(a, 0) the equation of a parabola
is y2= 4ax
The parametric form of the equation is x =
at2, y = 2at
The equation of the tangent at (x1, y1) is yy1
= 2a(x + x1)
Ellipse (see Figure 2.11)
SP
Eccentricity e = 33 < 1
The equation of an ellipse is 33 + 33 = 1a2 b
where b2= a2 (1 – e2
The equation of the tangent at (x1, y1) is
33 + 33 = 1
a2 b2
The parametric form of the equation of an
ellipse is x = a cos , y = b sin, where is
the eccentric angle
Hyperbola (see Figure 2.12)
SP
Eccentricity e = 33 > 1
The equation of a hyperbola is 33 – 33 = 1a2 b
where b2= a2(e2
Trang 1545 Fundamental dimensions and units
y axis
Focus S(a,0)
x axis
Fig 2.10 Parabola
D P
S(ae,0)
b
b
a a
x axis
y axis
Fig 2.11 Ellipse
y axis
a a
x axis
D
S
P S(ae,0)
Fig 2.12 Hyperbola
Trang 1646 Aeronautical Engineer’s Data Book
The parametric form of the equation is x =
a sec , y = b tan where s the eccenteric
angle
The equation of the tangent at (x1, y1) is
xx1 yy1
33 – 33 = 1
a2 b2
Sine Wave (see Figure 2.13)
y = a sin(bx + c)
y = a cos(bx + c') = a sin(bx + c) (where c = c'+
where a = �m2+ n�, c = tan–1 (n/m)
y axis
x axis c/b
a
2 π/b
0
Fig 2.13 Sine wave
Helix (see Figure 2.14)
A helix is a curve generated by a point moving
on a cylinder with the distance it transverses parallel to the axis of the cylinder being proportional to the angle of rotation about the axis:
x = a cos
y = a sin
z = k
where a = radius of cylinder, 2 πk = pitch
Trang 1747 Fundamental dimensions and units
a
z
y
x
Fig 2.14 Helix
2.9 Useful references and standards
For links to ‘The Reference Desk’ – a website containing over 6000 on-line units conversions
‘calculators’ – go to: www.flinthills.com/
~ramsdale/EngZone/refer.htm
United States Metric Association, go to: http://lamar.colostate.edu/~hillger/ This site contains links to over 20 units-related sites For guidance on correct units usage go to: http://lamar.colostate.edu/~hillger/correct.htm
Standards
1 ASTM/IEEE SI 10: 1997: Use of the SI
system of units (replaces ASTM E380 and
IEEE 268)
2 Taylor, B.N Guide for the use of the Inter
national System of units (SI): 1995 NIST
special publication No 8111
Trang 1848 Aeronautical Engineer’s Data Book
3 Federal Standard 376B: 1993: Preferred
Metric Units for general use by the Federal Government General Services Administra
tion, Washington DC, 20406
Trang 19Section 3
Symbols and notations
3.1 Parameters and constants
See Table 3.1
Table 3.1 Important parameters and constants
Planck’s constant (h)
Universal gas constant (R)
Stefan–Boltzmann constant ( )
Acceleration due to gravity (g)
Absolute zero
Volume of 1 kg mol of ideal
gas at 1 atm, 0°C
Avagadro’s number (N)
Speed of sound at sea level (a0)
Air pressure at sea level (p0)
6.6260755 10 –34 J s 8.314510 J/mol/K 5.67051 10 –8 W/m 2 K 4
9.80665 m/s 2
(32.17405 ft/s 2 ) –273.16°C (–459.688°F) 22.41 m 3
6.023 10 26 /kg mol 340.29 m/s (1116.44 ft/sec)
760 mmHg
= 1.01325 10 5 N/m 2
= 2116.22 lb/ft 2
Air temperature at sea level (T0
Air density at sea level ( 0 ) 1.22492 kg/m 3
slug/ft 3 ) Air dynamic viscosity at sea 1.4607 10 –5 m 2 /s
3.2 Weights of gases
See Table 3.2
Table 3.2 Weights of gases
Gas kg/m 3 lb/ft 3