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Natural Sciences Tripos Part IB

Mathematical Methods I

Michaelmas 2014

Dr Henrik Latter

i

2

Schedule

Vector calculus. Suffix notation. Contractions using ij and ijk. Re-

minder of vector products, grad, div, curl, 2 and their representationsusing suffix notation. Divergence theorem and Stokess theorem. Vector

differential operators in orthogonal curvilinear coordinates, e.g. cylindrical

and spherical polar coordinates. Jacobians. [6 lectures]

Partial differential equations. Linear second-order partial differential

equations; physical examples of occurrence, the method of separation of

variables (Cartesian coordinates only). [2]

Greens functions. Response to impulses, delta function (treated heuris-

tically), Greens functions for initial and boundary value problems. [3]

Fourier transform. Fourier transforms; relation to Fourier series, sim-

ple properties and examples, convolution theorem, correlation fnuctions,

Parsevals theorem and power spectra. [2]

Matrices. N -dimensional vector spaces, matrices, scalar product, trans-

formation of basis vectors. Eigenvalues and eigenvectors of a matrix; de-

generate case, stationary property of eigenvalues. Orthogonal and unitary

transformations. Quadratic & Hermitian forms, quadric surfaces. [5]

Elementary analysis. Idea of convergence and limits. O notation.

Statement of Taylors theorem with discussion of remainder. Convergence

of series; comparison and ratio tests. Power series of a complex variable;

circle of convergence. Analytic functions: CauchyRiemann equations,

rational functions and exp(z). Zeros, poles and essential singularities. [3]

Series solutions of ordinary differential equations. Homogeneous

equations; solution by series (without full discussion of logarithmic sin-

gularities), exemplified by Legendres equation. Classification of singular

points. Indicial equations and local behaviour of solutions near singular

points. [3]

3

Course websites

camtools.cam.ac.uk Maths : NSTIB 14 15

www.damtp.cam.ac.uk/user/hl278/NST

Assumed knowledge

I will assume familiarity with the following topics at the level of Course A

of Part IA Mathematics for Natural Sciences.

Algebra of complex numbers

Algebra of vectors (including scalar and vector products)

Algebra of matrices

Eigenvalues and eigenvectors of matrices

Taylor series and the geometric series

Calculus of functions of several variables

Line, surface and volume integrals

The Gaussian integral

First-order ordinary differential equations

Second-order linear ODEs with constant coefficients

Fourier series

4

Textbooks

The following textbooks are recommended. The first grew out of the NST

Maths course, so it will be particularly close to the lectures in places.

K. F. Riley, M. P. Hobson and S. J. Bence (2006). MathematicalMethods for Physics and Engineering, 3rd edition. Cambridge Univer-

sity Press

G. Arfken and H. J. Weber (2005). Mathematical Methods for Physi-cists, 6th edition. Academic Press

Questions

Please ask questions in lecture, especially brief ones (typos, jargon)

Longer questions can be e-mailed to me: [email protected] Turnaroundtime: approximately 1-3 days

1 VECTOR CALCULUS 1

1 Vector calculus

1.1 Motivation

Scientific quantities are of different kinds:

scalars have only magnitude (and sign), e.g. mass, electric charge,energy,

temperature

vectors have magnitude and direction, e.g. velocity, magnetic field,temperature gradient

A field is a quantity that depends continuously on position (and possibly

on time). Examples:

air pressure in this room (scalar field)

electric field in this room (vector field)

Vector calculus is concerned with scalar and vector fields. The spatial vari-

ation of fields is described by vector differential operators, which appear

in the partial differential equations governing the fields.

Vector calculus is most easily done in Cartesian coordinates, but other

systems (curvilinear coordinates) are better suited for many problems be-

cause of symmetries or boundary conditions.

1.2 Suffix notation and Cartesian coordinates

1.2.1 Three-dimensional Euclidean space

This is a close approximation to our physical space:

points are the elements of the space

vectors are translatable, directed line segments

1 VECTOR CALCULUS 2

Euclidean means that lengths and angles obey the classical results ofgeometry

Points and vectors have a geometrical existence without reference to any

coordinate system. For definite calculations, however, we must introduce

coordinates.

Cartesian coordinates:

measured with respect to an origin O and a system of orthogonal axesOxyz

points have coordinates (x, y, z) = (x1, x2, x3)

unit vectors (ex, ey, ez) = (e1, e2, e3) in the three coordinate direc-tions, also called (, , k) or (x, y, z)

The position vector of a point P is

OP = r = exx+ eyy + ezz =

3i=1

eixi

1.2.2 Properties of the Cartesian unit vectors

The unit vectors form a basis for the space. Any vector a belonging to

the space can be written uniquely in the form

a = exax + eyay + ezaz =3

i=1

eiai

where ai are the Cartesian components of the vector a.

The basis is orthonormal (orthogonal and normalized):

e1 e1 = e2 e2 = e3 e3 = 1

e1 e2 = e1 e3 = e2 e3 = 0

1 VECTOR CALCULUS 3

and right-handed :

[e1, e2, e3] e1 e2 e3 = +1

This means that

e1 e2 = e3 e2 e3 = e1 e3 e1 = e2

The choice of basis is not unique. Two different bases, if both orthonor-

mal and right-handed, are simply related by a rotation. The Cartesian

components of a vector are different with respect to two different bases.

1.2.3 Suffix notation

xi for a coordinate, ai (e.g.) for a vector component, ei for a unitvector

in three-dimensional space the symbolic index i can have the values 1,2 or 3

quantities (tensors) with more than one index, such as aij or bijk, canalso be defined

Scalar and vector products can be evaluated in the following way:

a b = (e1a1 + e2a2 + e3a3) (e1b1 + e2b2 + e3b3)

= a1b1 + a2b2 + a3b3 =3

i=1

aibi

a b = (e1a1 + e2a2 + e3a3) (e1b1 + e2b2 + e3b3)

= e1(a2b3 a3b2) + e2(a3b1 a1b3) + e3(a1b2 a2b1)

=

e1 e2 e3

a1 a2 a3

b1 b2 b3

1 VECTOR CALCULUS 4

1.2.4 Delta and epsilon

Suffix notation is made easier by defining two symbols. The Kronecker

delta symbol is

ij =

1 if i = j0 otherwiseIn detail:

11 = 22 = 33 = 1

all others, e.g. 12 = 0

ij gives the components of the unit matrix. It is symmetric :

ji = ij

and has the substitution property :

3j=1

ijaj = ai (in matrix notation, 1a = a)

The Levi-Civita permutation symbol is

ijk =

1 if (i, j, k) is an even permutation of (1, 2, 3)

1 if (i, j, k) is an odd permutation of (1, 2, 3)

0 otherwise

In detail:

123 = 231 = 312 = 1

132 = 213 = 321 = 1

all others, e.g. 112 = 0

1 VECTOR CALCULUS 5

An even (odd) permutation is one consisting of an even (odd) number of

transpositions (interchanges of two neighbouring objects). Therefore ijk

is totally antisymmetric : it changes sign when any two indices are inter-

changed, e.g. jik = ijk. It arises in vector products and determinants.

ijk has three indices because the space has three dimensions. The even

permutations of (1, 2, 3) are (1, 2, 3), (2, 3, 1) and (3, 1, 2), which also

happen to be the cyclic permutations. So ijk = jki = kij.

Then we can write

a b =3

i=1

aibi =3

i=1

3j=1

ijaibj

a b =

e1 e2 e3

a1 a2 a3

b1 b2 b3

=3

i=1

3j=1

3k=1

ijkeiajbk

1.2.5 Einstein summation convention

The summation sign is conventionally omitted in expressions of this type.

It is implicit that a repeated index is to be summed over. Thus

a b = aibi

and

a b = ijkeiajbk or (a b)i = ijkajbk

The repeated index should occur exactly twice in any term. Examples of

invalid notation are:

a a = a2i (should be aiai)

(a b)(c d) = aibicidi (should be aibicjdj)

The repeated index is a dummy index and can be relabelled at will. Other

indices in an expression are free indices. The free indices in each term in

an equation must agree.

1 VECTOR CALCULUS 6

Examples:

a = b ai = bi

a = b c ai = ijkbjck|a|2 = b c aiai = bici = bjcja = (b c)d+ e f ai = bjcjdi + ijkejfk

Contraction is an operation by which we set one free index equal to an-

other, so that it is summed over. For example, the contraction of aij is

aii. Contraction is equivalent to multiplication by a Kronecker delta:

aijij = a11 + a22 + a33 = aii

The contraction of ij is ii = 1 + 1 + 1 = 3 (in three dimensions).

If the summation convention is not being used, this should be noted

explicitly.

1.2.6 Matrices and suffix notation

Matrix (A) times vector (x):

y = Ax yi = Aijxj

Matrix times matrix:

A = BC Aij = BikCkj

Transpose of a matrix:

(AT)ij = Aji

Trace of a matrix:

trA = Aii

Determinant of a (3 3) matrix:

detA = ijkA1iA2jA3k

(or many equivalent expressions).