### Mathematical Methods For Physicists – Lecture 1 – 02.10.07

**Functions of More Than One Variable**

e.g. Given region of xy plane 0≤y≤1, 0≤x≤1 - call this M (see diagram 1)

We say a ral function of real variables x,y is defined in M if a rule is given according to which exactly one real number is assigned to each point (x, y) of M is called the **domain of definition of the function**.

We write z = f(x, y)

Ex 1 z = x^{2}y defined for whole xy plane – domain is xy plane.

Ex 2 z = √1-x^{2}-y^{2} domain is all points (x, y) such that 1 – x^{2} – y^{2} ≥ 0

i.e. inside and on x^{2} + y^{2} = 1

Ex 3 z = √(1-x_{1}^{2}-x_{2}^{2}-…-x_{N}^{2}) domain is x_{1}^{2} + x_{2}^{2} + … + x_{N}^{2} ≥ 1

Closed since inside boundary.

**Viewing A Function**

Think of z as a height about the xy plane (see diagram 2)

**Composite Functions**

z is a function of two functions, each of which is a function of xy

z = h(f(x,y)g(x,y))

We say we have a composite function of U = f(x,y) and V = g(x,y).

Such that z = h(u, v)

u, v defined over xy plane.

We require to define z over a set N such that U, V Ç N

Ç in this context is used to denote 'contains', since the blog doesn't support the correct symbol.

Ex 1 z = (1+x^{2}+y^{2})^{sin xy} → composite function of U = 1+x^{2}+y^{2}

V = sin xy

Write z = U^{V}

M à whole of xy plane

N could be xy such that 0 ≤ xy ≤π gives > 0

or π ≤ xy ≤ 2π gives < 0

Ex 2

z = √(1-x^{2}sinx) composite of u = x^{2}, v = sin x

z = √(1-UV)

only has domain for UV ≥ 1 i.e. only on one side of hyperbola UV = 1 and on it

(See diagram 3)

**The Distance Between Two Points**

In Cartesian coordinates the dsiance between (x_{1}, y_{1}) and (x_{2}, y_{2}) is given by

d = √((x_{1}-x_{2})^{2} + (y_{1}-y_{2})^{2})

In n dimension for 2 points (x_{1}, x_{2, }…, x_{N}) and (y_{1}, y_{2}, …, y_{N})

d = √((x_{1}-y_{1})^{2} + (x_{2}-y_{2})^{2} + … + (x_{N}-y_{N})^{2})

**The ****δ**** Neighbourhood Of A Point P**

The set of all point whose distance from P is <δ

e.g. in xy plane δ neighbourhood of a point (x_{0}, y_{0}) is all points within circle given by

(x-x_{0})^{2 }+ (y-y_{0})^{2} + δ^{2}

The limits, A, of a function z = f(x, y) at P(x_{0}, y_{0}) **UNINTELLIGIBLE NONESENSE!**

We say z has a linit A at P to every **UNINTELLIGIBLE NONESENSE!**

Then exists a δ > 0.

Such that for all points in the d neighbourhood **UNINTELLIGIBLE NONESENSE!**

**Intuitive Meaning**

f has a limit, A, at P if f is sufficiently close to A. For all points sufficiently close to A **UNINTELLIGIBLE NONESENSE!**

**Continuity**

f is continuous at (x_{0}, y_{0}) if

1) it is defined at this point

2) If, corresponding to an area ε > 0 there exists a δ > 0.

We have |f(x, y) - f(x_{0}, y_{0})| < ε

**Working Principle**

A) Can say that f(x, y) is defined at (x_{0}, y_{0}) then it is continuous at that point if, and only if,

lim f(x, y) = f(x_{0}, y_{0})

(x, y) à (x_{0}, y_{0})

We say it is continuous in M.

Limits is f(x, y) à A and g(x, y) à B

k f(x,y) = kA

lim f(x,y) ± g(x,y) = A±B

(x, y) à (x_{0}, y_{0}) f(x,y)g(x,y) = AB

^{f(x,y)}/_{g(x,y)} = ^{A}/_{B} g ¹ 0

**Theorems Without Proof**

1) A continuous function of a continuous function is continuous

2) If f(x,y) is continuous in a region R and if (x_{1}, y_{1}) and (x_{2}, y_{2}) are any two points of R then f(x, y) then in R every value between f(x, y) and f(x_{0}, y_{0}).

3) A function hat is continuous in a **SOMETHING** closed region R’, takes on a greatest value at least at one point (x_{0}, y_{0}) Ç R’

i.e. f(x_{0}, y_{0}) ≥ f(x, y) for all points (x, y) Ç R’ at least value at limit of the point (x, y) Ç R.