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Biologists are transforming the proteinmaking instructions of Escherichiacoli.
C. Bickel/
zero (corresponding to local maxima, minima or saddle points) and are given by the conditions
f
f
=
= 0.
x
y
3.1.9
Taylor Series
We can approximate the behaviour of the function at a point using knowledge of its derivatives.
For example, suppose a car is
1
Basic Skills
This document contains notes on basic mathematics. There are links to the corresponding Leeds
University Library [email protected] page, in which there are subject notes, videos and examples.
If you require more in-depth explanations of these co
4.8
Polar Form
A useful representation of a complex number is in polar coordinates.
For example: the complex number:
by
2 + i as shown in Figure 9 can also be represented
(r, ) , where r is the magnitude of z (the distance from 0) and is the angle with th
imaginary
y
real
x
Figure 8: An argand diagram showing the magnitude |z| =
4.7
z z of the complex number z = x + iy.
Division
It is not immediately obvious how to divide two complex numbers z1 and z2 . However, we do
know how to divide a complex number by
5.1
Plus/Minus Notation
value standard deviation units.
Errors are often quoted in the form:
For example:
100 1 kg/cm3 ,
=
where 1 here is the standard deviation. With this
terminology, it is possible that is equal to 98 or 103 kg/cm3 , although very unli
Therefore:
Z
f = cos xx
cos xx + sin x + C
=
Check!
1( cos x) dx
d
(x cos x + sin x)
dx
=
cos x + x sin x + cos x
=
x sin x
2. By substitution
For example, suppose we want to find
Z
x cos (x2 + 1) dx
Write
u = x2 + 1,
so that
du
dx
= 2x or rearranging
Z
b
f (x) dx
=
a
b
F (x) a
=
F (b) F (a)
(28)
where F 0 (x) = f (x)
That is, F is the anti-derivative of f .
Z
2
h 1 i2
x2 dx =
x3
3
1
1
d 1 3
= x2 .
x
since,
dx 3
=
e.g.
1
(8 1)
3
=
7
3
An indefinite integral is an integral without limits and gives a fun
ample: the top left element in matrix A, equal to 1, is in row 1 and column 1 and can be
labelled as element a11 ; the element in the 2nd column of row 1, equal to 3, is labelled as a12 .
A general element aij is located in row i and column j (see equatio
For example:
1. How do we differentiate the function f (t) = sin 2t ?
We know how to differentiate f = sin u, so lets define u = 2t. Then we simply need to
assemble the ingredients for the chain rule:
du
dt
= 2 and
df
du
= cos u. It then follows that
df
d
2.2
Linear Systems
Wolfram link
Video link
(http:/mathworld.wolfram.com/LinearSystemofEquations.html)
(http:/www.youtube.com/watch?v=ZK3O402wf1c)
A linear system of equations such as
5x + 3y + z = 3
2x 10y 3z = 1
4y + 5z = 7
(17)
can be written as
1 x
5
4
Complex Numbers
Library link
Wolfram link
Video link
(http:/library.leeds.ac.uk/tutorials/maths-solutions/pages/complex numbers/ )
(http:/mathworld.wolfram.com/ComplexNumber.html)
(http:/ocw.mit.edu/resources/res-18-008-calculus-revisited-complex-variab
f (1, ) = 0;
f
= y cos(xy)
=
x
(1,)
f
= 1
y (1,)
Hence, f (1 + h, + k) = h k,
3.2
Integration
Wolfram link
to 1st order.
(http:/mathworld.wolfram.com/Integral.html)
The integral of a function f between x = a and x = b is the area under the curve of f :
distance
t
t t+h
Figure 2
For example
Suppose f (t) = t2 , then
f (t + h) f (t)
(t + h)2 t2
=
h
h
2th + h2
=
h
= 2t + h
therefore:
f (t + h) f (t)
2t
h
as h 0
Hence the gradient of f at any point t is 2t.
3.1.1
Notation
The gradient or derivative of a fu
Then we can check that
V V 1
1 1 1 1 1 0 1 1
2 2
1 1
0 3
1 1
1 1 3 1 1
2
1 3
1 1
1 2
=
A
2 1
=
=
=
Why is this useful?
Example:
A8 =
What is A8 ? Using the matrix diagonalisation,
V V 1
V V 1
V V 1
V V 1
V V 1
V V 1
V V 1
V V 1
= V V 1 V V 1 V V 1 V
Figure 12
gives the likelihood of x is:
f (x) =
1
2 2
e
(x)2
2 2
,
(46)
which is normalised such that
Z
f (x) dx = 1.
(47)
5.5
Central limit theorem
Normal distributions often arise out of other non-normal distributions. If x1 , x2 , etc . are
identical v
imaginary
i
-1
1
real
-i
Figure 6
then z is purely real; if x = 0, then z is purely imaginary.
4.3
Complex Plane
We can plot a complex number in an x y domain called the complex plane or the Argand
Diagram. For example, z = x + iy is displayed in the Arga
imaginary
real
Figure 11
In Figure 11, = +
4
=
5
.
4
NB you need to take care with finding the phase , since the inverse sine and cosine functions may give the correct answer in the wrong range. i.e. your calculator will tell you that
1
sin1 =
4
2
and
tan
4.10
Application to waves
Some links about waves as a refresher:
Movie link
Sine and cosine waves
(http:/videos.kightleys.com/Science/Maths/23131008 CsD3fs/1880848370
VMGSWd3#!i=1880848370&k=VMGSWd3)
Movie link
Superposition of waves
(http:/www.acs.psu.ed
i.e. if n = 2
1 0
I=
0 1
(13)
This is a special case of a diagonal matrix possessing non-zero entries only on its diagonal e.g.
2 0 0
0 3 0
0 0 1
If A is a square n n matrix, then the identity matrix Inn has the special property that:
AI = IA = A
(14)
then,
a11 + b11 a12 + b12
A+B =
a21 + b21 a22 + b22
2.1.4
(5)
Subtraction
Similar to addition, corresponding elements in A and B are subtracted from each other:
a11 b11 a12 b12
AB =
a21 b21 a22 b22
2.1.5
(6)
Multiplication by a scalar
If is a number (
Standard Results
R
f (x)
1
x+C
xn
3.2.1
f (x)dx
1
xn+1
n+1
+C
cos x
sin x + C
sin x
cos x + C
ex
ex + C
1
x
ln |x| + C
Finding Integrals
1. By parts
We have already seen that: (uv)0 = u0 v + v 0 u.
If we integrate this:
Z
Z
0
(uv) dx =
u0 v + v 0 u
Z
uv
We can also have higher derivatives. Consider the gradient of a gradient function. If f represents distance, then f 0 is the speed and f 00 is the acceleration.
3.1.2
Standard Results
f
f0
1
0
t
1
tn
ntn1
tm
mt(m+1)
sin t
cos t
cos t
sin t
et
et
ln t
1
t
In order that is unique, arg(z) is often used.
4.9
Exponential Notation
It turns out that:
z = r (cos + i sin )
can also be written as
z = rei ,
(35)
(36)
where
ei = cos + i sin
(37)
This makes it easy to multiply and divide in polar form:
z1 z2
=
r1 ei1
imaginary
i
real
-1
1
Figure 4
imaginary
-1
1
real
-i
Figure 5
1, i, i2 , i3 , i4 , = 1, i, 1, i, 1, i, 1, i, 1, . . . as depicted below in Figure 6.
4.2
Definition
A complex number z is written (x, y) or x + iy where x is the real part of z and y is the
imaginary
z
y
real
x
Figure 7: An argand diagram showing the complex plane. The complex number z has real part x and
complex part y.
4.5
Multiplication
For two complex numbers z1 and z2 , find their product by multiplying out in full.
For example:
(5 + 2i
2. How many bolts are needed for Alexs car parts?
(0 3) + (1 1) + (4 2) = 9.
3. How many screws are needed for Peters car parts?
(1 4) + (0 8) + (2 4) = 6.
Or we can use matrix multiplication to get all four scenarios:
3 4
7 6
1 0 2
=
AB =
1
8
0
or
1 0 0
2 3 4
5 6 7
3 4
= 1
6 7
0 0
2
6 7
0 0
+
5
3 4
= 3
Do whichever is easier!
2.3.1
Using determinants to invert a 2 2 matrix
The determinant
can be
used in finding the inverse of a 2 2 matrix.
a b
For A =
, the inverse can be found