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Introduction

Let us consider the concepts of limits, continuity and differentiability of functions of two (or more) real variables. Functions with two (or more) independent variables are common in science. Their derivatives are interesting and their integrals lead to a variety of applications.

The Cartesian product A × B of sets A and B is the collection of all ordered pairs (a, b) with a ∈ A and b ∈ B. If A ⊆ R and B ⊆ R, A × B is a subset of R × R, i.e., R2 whose elements are considered to be points in the 2-D plane. If (a, b) ∈ R2, it is associated with a unique point in the 2-D plane with Cartesian co-ordinates (a, b).

If we denote A × B = E, so that E ⊂ R2, then the function f : E → R is termed a real valued function of two variables. For (x, y) ∈ E, it's image under f is denoted by f(x, y) or 'z' (a real number). Henceforth, a function of two variables means a real valued function of two real variables.

Basic Concepts

Circular neighbourhood:
Let (a, b) ∈ R2. For a positive number δ, the set {(x, y) ∈ R2 : < δ} is called the δ-neighbourhood of (a, b) or as the circular neighbourhood of (a, b) with radius δ. Refer adjacent figure.

Rectangular neighbourhood:
Let (a, b) ∈ R2. For two positive numbers δ1 and δ2, the set {(x, y) ∈ R2 : |x – a| < δ1 and |y – b| < δ2} is called the rectangular neighbourhood of (a, b). Refer adjacent figure.

Note that every rectangular neighbourhood of (a, b) contains a circular neighbourhood of (a, b). The vice–versa is also true.

Simultaneous limit:
A function f : E → R is said to tend to a limit 'l' as (x, y) approaches (a, b).
Symbolically .

Note that the difference between 'l' and the values taken by the function has to be made arbitrarily small for all points close to (a, b). In other words, ⇔ for a given ∈ > 0, there exists a δ > 0 such that, for all (x, y) ∈ E with 0 < < δ, we have |f(x, y) – l| < ∈. Then 'l' is called the simultaneous limit of f as (x, y) nears (a, b).

Repeated limit:
Let f : E → R be a function of two variables, where E ⊂ R2.
For any x ∈ R, let Ex = {y ∈ R : (x, y) ∈ E} so that Ex ⊂ R.
For a given 'x', f is considered as a real-valued function on Ex.
If for each 'x', the following two limits exist:
(i) and
(ii)
then 'α' is called a repeated limit.
It is denoted by

Similarly, we can also define the second repeated limit .

Note:
i) If the simultaneous limit of a function exists, then the two repeated limits also exist. And all the three limits have the same value.
ii) But if the repeated limits exist and are equal, the simultaneous limit may or may not exist.

Continuity:
The function f : E → R is said to be continuous at (a, b) ∈ E, if exists and is equal to f(a, b).
That is, f : E → R is continuous at (a, b) ∈ E if, for every ∈ > 0, there exists a δ > 0 such that for all (x, y) ∈ E, with 0 < < δ, we have |f(x, y) – f(a, b)| < ∈ or = f(a, b).

Partial Derivatives

Partial derivatives – first order: In the function f(x, y), if 'y' is held constant and the function is differentiated with respect to 'x', the resulting derivative is called the partial derivative of f with respect to 'x'. It is denoted by or fx.
Thus = fx = , if the limit exists.

Similarly, the partial derivative of f with respect to 'y' is: = fy = , if the limit exists.

Thus fx and fy are called first order partial derivatives of 'f' with respect to 'x' and 'y' respectively.

Ex: If f(x, y) = xey + yex, then find fx and fy.
Sol: fx = (1)ey + y(ex) = ey + yex
fy = x(ey) + (1)ex = xey + ex

Partial derivatives – second order: and are themselves functions of 'x' and 'y'. In turn, they may possess partial derivatives. These are called second order partial derivatives of 'f'.

These are also represented by fxx and fyy. The condition, of course, is that the two limits should exist.

Ex: If f(x, y) = x2 – y2, then find fxx and fyy.
Sol:
fx = 2x.(1) – 0 = 2x ; fy = 0 – 2y.(1) = –2y
fxx = 2(1) = 2 ; fyy = –2(1) = –2

There are also mixed second order partial derivatives.
These are defined as:

Differentiation of Composite Functions

Stated below are 3 rules for differentiation of a composite function of two variables.

i) z is a function of u, i.e., z = f(u)
u is a function of x and y, i.e., u = g(x, y)
Then
Ex: If z = cos u and u = 2x2 – 4y2, then find and .
Sol:
= . = –sin u . 4x
= –4x . sin (2x2 – 4y2)
= . = –sin u . (–8y)
= 8y . sin (2x2 – 4y2)
ii) z is a function of x and y, i.e., z = f(x, y)
x and y are functions of 't', i.e., x = g(t), y = h(t)
Then
Ex: If z = ex + 2y, x = at2 and y = 2at, then find .
Sol:
= . + .
= ex + 2y ; = 2ex + 2y
= 2at ; = 2a
Hence, = (ex + 2y)(2at) + (2ex + 2y)(2a)
= 2aex + 2y(t + 2)
iii) z is a function of u and v, i.e., z = f(u, v)
u and v are functions of x and y, i.e., u = g(x, y), v = h(x, y).
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