Engineering 1B (H1034)
Week 6
19.3 - Second Order Differential Equations with constant coefficients
1. For a homogeneous differential equation of the form ay 00 + by 0 + cy = 0 :
Know how to use the auxiliary equation to find the values of k that satisfy
Engineering 1B (H1034)
Week 10
1
28.1 - Background to vector calculus
Make sure you are secure in using partial differentiation with two or more variables
(see Eng 1A notes), and in the concept of vectors and partial vector differentiation.
You should k
Engineering 1B (H1034)
Week 5
1. 19.1 - Differential Equations (d.e.s)
You should:
(a) Understand what is meant by a differential equation, its degree, and which variables represent the unknown (dependent) function and the given (independent)
2
t
function
Engineering 1B (H1034)
Week 2
1
15.2 - Calculating centres of mass
Know the meaning of centre of mass and the definition of moment.
Learn and know how to use the formula for calculating the centre of mass of individual
masses located on an axis, for exa
Engineering 1B (H1034)
Week 1
1
14.4 - Lengths of curves and surfaces of revolution
(From the previous HELM section) Learn and know how to use the volume of revolution formula for y = f (x):
Z b
V =
(f (x)2 dx.
a
Learn and know how to use the arc length
Engineering 1B (H1034)
Week 4
1. 16.5 - Maclaurin and Taylor Series
You should know:
(a) the general Taylor Series formula,
(b) the difference between a general Maclaurin and a general Taylor series formula
(the Maclaurin formula is exactly the same as th
Engineering 1B (H1034)
Week 9
1
27.3 - Volume integrals
You should know how to evaluate triple integrals in Cartesian, Cylindrical and Spherical coordinates.
Their volume integrals are, respectively:
Z bZ dZ f
Z 2Z aZ c
Z Z aZ
dzdydx,
rdzdrd,
a
c
e
0
0
b
Engineering 1B (H1034)
Week 11
1
29.1 - Line integrals
You should be able to calculate line integrals over scalar fields of the form
r
2
p
dy
2
2
where ds may be written as ds = dx + dy , or ds = 1 + dx
.
R
C
F (x, y)ds,
R
You should be able to calculate
Engineering 1B (H1034)
Week 3
1. 16.1 - Sequences and Series
You should know:
(a) the difference between a sequence and a series,
(b) the notation for the value of a term in a sequence or a series, for example an is
the value of the nth term,
P
(c) the su
Differential Vector
Calculus
28.2
Introduction
A vector field or a scalar field can be differentiated with respect to position in three ways to produce
another vector field or scalar field. This Section studies the three derivatives, that is: (i) the gra
Contents
27
Multiple Integration
27.1 Introduction to Surface Integrals
2
27.2 Multiple Integrals over Non-rectangular Regions
20
27.3 Volume Integrals
41
27.4 Changing Coordinates
66
Learning outcomes
In this Workbook you will learn to integrate a functi
Contents
28
Differential
Vector Calculus
28.1 Background to Vector Calculus
2
28.2 Differential Vector Calculus
17
28.3 Orthogonal Curvilinear Coordinates
37
Learning outcomes
In this Workbook you will learn about scalar and vector fields and how physical
Multiple Integrals
over Non-rectangular
Regions
27.2
Introduction
In the previous Section we saw how to evaluate double integrals over simple rectangular regions. We
now see how to extend this to non-rectangular regions.
In this Section we introduce funct
Contents
19
Differential Equations
19.1 Modelling with Differential Equations
2
19.2 First Order Differential Equations
11
19.3 Second Order Differential Equations
30
19.4 Applications of Differential Equations
51
Learning outcomes
In this Workbook you wi
Contents
29
Integral Vector
Calculus
29.1 Line Integrals Involving Vectors
2
29.2 Surface and Volume Integrals
34
29.3 Integral Vector Theorems
54
Learning outcomes
In this Workbook you will learn how to integrate functions involving vectors. You will lea
Maclaurin and Taylor
Series
16.5
Introduction
In this Section we examine how functions may be expressed in terms of power series. This is an
extremely useful way of expressing a function since (as we shall see) we can then replace complicated
functions in
Second Order
Differential Equations
19.3
Introduction
In this Section we start to learn how to solve second order differential equations of a particular type:
those that are linear and have constant coefficients. Such equations are used widely in the mode
Integral Vector
Theorems
29.3
Introduction
Various theorems exist equating integrals involving vectors. You have already met the fundamental
theorem of line integrals. Those involving line, surface and volume integrals are introduced here.
They are the mu
Applications of
Eigenvalues and
Eigenvectors
22.2
Introduction
Many applications of matrices in both engineering and science utilize eigenvalues and, sometimes,
eigenvectors. Control theory, vibration analysis, electric circuits, advanced dynamics and qua
The Binomial Series
16.3
Introduction
In this Section we examine an important example of an infinite series, the binomial series:
p(p 1) 2 p(p 1)(p 2) 3
x +
x +
2!
3!
We show that this series is only convergent if |x| < 1 and that in this case the series
Changing Coordinates
27.4
Introduction
We have seen how changing the variable of integration of a single integral or changing the
coordinate system for multiple integrals can make integrals easier to evaluate. In this Section we
introduce the Jacobian. Th
Volume Integrals
27.3
Introduction
In the previous two Sections, surface integrals (or double integrals) were introduced i.e. functions
were integrated with respect to one variable and then with respect to another variable. It is often
useful in engineer
Moment of Inertia
15.3
Introduction
In this Section we show how integration is used to calculate moments of inertia. These are essential
for an understanding of the dynamics of rotating bodies such as flywheels.
Prerequisites
Before starting this Section
Contents
22
Eigenvalues and
Eigenvectors
22.1 Basic Concepts
2
22.2 Applications of Eigenvalues and Eigenvectors
18
22.3 Repeated Eigenvalues and Symmetric Matrices
30
22.4 Numerical Determination of Eigenvalues and Eigenvectors
46
Learning outcomes
In th
16.2
Infinite Series
Introduction
We extend the concept of a finite series, met in Section 16.1, to the situation in which the number
of terms increase without bound. We define what is meant by an infinite series being convergent
by considering the parti
Power Series
16.4
Introduction
In this Section we consider power series. These are examples of infinite series where each term
contains a variable, x, raised to a positive integer power. We use the ratio test to obtain the radius
of convergence R, of the
Contents
16
Sequences and Series
16.1 Sequences and Series
2
16.2 Infinite Series
13
16.3 The Binomial Series
26
16.4 Power Series
32
16.5 Maclaurin and Taylor Series
40
Learning outcomes
In this Workbook you will learn about sequences and series. You wil
Calculating
Centres of Mass
15.2
Introduction
In this Section we show how the idea of integration as the limit of a sum can be used to find the
centre of mass of an object such as a thin plate (like a sheet of metal). Such a plate is also known
as a lami
Surface and
Volume Integrals
29.2
Introduction
A vector or scalar field - including one formed from a vector derivative (div, grad or curl) - can be
integrated over a surface or volume. This Section shows how to carry out such operations.
Prerequisites
Be
Contents
15
Applications of
Integration 2
15.1 Integration of Vectors
2
15.2 Calculating Centres of Mass
5
15.3 Moment of Inertia
24
Learning outcomes
In this Workbook you will learn to interpret an integral as the limit of a sum. You will learn
how to ap