October 04, 2007

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 = x2y defined for whole xy plane – domain is xy plane.

Ex 2    z = √1-x2-y2    domain is all points (x, y) such that       1 – x2 – y2 ≥ 0

                                   i.e. inside and on x2 + y2 = 1

Ex 3    z = √(1-x12-x22-…-xN2)           domain is         x12 + x22 + … + xN2 ≥ 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+x2+y2)sin xy         composite function of          U = 1+x2+y2

                                                                                               V = sin xy

Write z = UV

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-x2sinx) composite of u = x2, 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 (x1, y1) and (x2, y2) is given by

           d = ((x1-x2)2 + (y1-y2)2)

In n dimension for 2 points (x1, x2, …, xN) and (y1, y2, …, yN)

           d = ((x1-y1)2 + (x2-y2)2 + … + (xN-yN)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 (x0, y0) is all points within circle given by

           (x-x0)2 + (y-y0)2 + δ2

The limits, A, of a function z = f(x, y) at P(x0, y0) 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 (x0, y0) 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(x0, y0)| < ε

Working Principle

A) Can say that f(x, y) is defined at (x0, y0) then it is continuous at that point if, and only if,

           lim       f(x, y) = f(x0, y0)

(x, y) à (x0, y0)

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 (x0, y0)           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 (x1, y1) and (x2, y2) are any two points of R then f(x, y) then in R every value between f(x, y) and f(x0, y0).

3)     A function hat is continuous in a SOMETHING closed region R’, takes on a greatest value at least at one point (x0, y0Ç R’

i.e. f(x0, y0) f(x, y) for all points (x, y) Ç R’ at least value at limit of the point (x, y) Ç R.


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