Function of Several Variables, Lecture notes of Calculus

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Functions of Several VariablesFunctions of Several Variables Dr. Hina Dutt [email protected]

So far, we have dealt with the calculus of

functions of a single variable.

However, in the real world, physical quantities

often depend on two or more variables.

The temperature T at a point on the surface

of the earth at any given time depends on

the longitude x and latitude y of the point.

 (^) We can think of T as being a function of the two variables x and y , or as a function of the pair ( x , y ).  (^) We indicate this functional dependence by writing: T = f ( x, y )

FUNCTIONS OF TWO VARIABLES

The volume V of a circular cylinder depends on its radius r and its height h.

 In fact, we know that V = πr 2 h.

 We say that V is a function of r and h.

 We write V ( r , h ) = πr 2 h.

FUNCTIONS OF TWO VARIABLES

We often write z = f ( x , y ) to make explicit the

value taken on by f at the general point ( x, y ).

 (^) The variables x and y are independent variables.  (^) z is the dependent variable.  (^) Compare this with the notation y = f ( x ) for functions of a single variable.

FUNCTIONS OF TWO VARIABLES

FUNCTIONS OF TWO VARIABLES

For each of the following functions,

evaluate f (3, 2) and find the domain.

a.

b.

FUNCTIONS OF TWO VARIABLES Example 1 1 ( , ) 1 x y f x y x     2 f ( , x y )  x ln( y  x )

 The expression for f makes sense if

the denominator is not 0 and the quantity

under the square root sign is nonnegative.

 So, the domain of f is:

D = {( x , y ) | x + y + 1 ≥ 0, x ≠ 1}

FUNCTIONS OF TWO VARIABLES Example 1 a 3 2 1 6 (3, 2) 3 1 2 f     

FUNCTIONS OF TWO VARIABLES Example 1 b

This is the set of points to the left of the parabola x = y 2 . FUNCTIONS OF TWO VARIABLES Example 1 b

In regions with severe winter weather,

the wind-chill index is often used to

describe the apparent severity of the cold.

 (^) This index W is a subjective temperature that depends on the actual temperature T and the wind speed v.  (^) So, W is a function of T and v , and we can write: W = f ( T , v ) FUNCTIONS OF TWO VARIABLES Example 2

The following table records values of W

compiled by the NOAA National Weather

Service of the US and the Meteorological

Service of Canada.

FUNCTIONS OF TWO VARIABLES Example 2

For instance, the table shows that, if

the temperature is –5°C and the wind speed

is 50 km/h, then subjectively it would feel

as cold as a temperature of about –15°C

with no wind.

 (^) Therefore, f (–5, 50) = – FUNCTIONS OF TWO VARIABLES Example 2

Find the domain and range of:

 The domain of g is:

D = {( x , y )| 9 – x^2 – y^2 ≥ 0}

= {( x , y )| x^2 + y^2 ≤ 9}

FUNCTIONS OF TWO VARIABLES

g x y ( , )  9  x  y Example 3