Contents [PD.1] [PD.2] [PD.3] [PD.4] [PD.5] [PD.6] [PD.7] [PD.8]
In this section, we will work with two variable systems. An example of a two variable system would be the following: At any longitude (first variable) and any latitude (second variable) there is a given temperature. Two variable systems work the same way in that two variables are chosen (longitude and latitude), then these variables are put in a equation, and the result of the equation is the third number (temperature). The formal definition is given below:
Definition:
of two variables is a rule that assigns to each ordered pair
of real numbers
in a set
a unique real number denoted
by
.
The set
is the domain of
and its range is the set of values that
takes on, that is,
We often write
to make explicit the value taken on by
at the general
.
The variables
and
are independent variables, that is that they can be any number within the
domain. The
value is the dependant variable, this means that this variable will depend on
the values of
and
The following are the graphs of multi variable functions.
Definition:
is a function of two variables with domain
,
then the graph of
is the set of all points
in
such that
and
is in
Example 1
f(x,y)=2+x-2y
You will notice that in this example if you put in a number for
and a number for
one will get a certain number out of the equation. The numbers x and y are
where the third number will be placed in the xy-plane, and the value of the
third number will be how far away from the xy-plane the point is placed. An
example that could be used would be the
and
.
If we plug those values into the equation we get a value of 1. Therefore,
the point will lie
unit above the point of
on the xy-plane.
This example is called an linear function. The graph of such a function has
the equation
,
or
,
so it is a plane. In much the same way that linear functions of one variable
are important in single-variable calculus, we will see that linear functions
of two variables play a central role in multivariable calculus.
Example 2
A way of visualizing two variable graphs is to draw out the graphs level curves. A level curve is points of consistent elevation that are joined together and then put on a 2-D graph.
Definition:
of two variables are the curves with equations
(x,y
)= k, where
is a constant (in the range of
A level curve
is the set of all points in the domain of
at which
takes on a given value
In other words, it shows where the graph of
has a height
.
This height is then projected onto the xy-plane in the form of a curve.
A function of three or more variables,
,
is a rule that assigns to each ordered triple
(
)
in a domain
a unique real number denoted by
.
For instance, the temperature T at a point on the surface of the earth depends
of the longitude
and the latitude
of the point and on the time
,
so we could write
It's vary difficult to visualize a function
of three variables by its graph, since that would lie in a four-dimensional
space. However, we do gain some insight into
by examining its level surfaces, which are the surfaces with equations
,
where
is a constant. If the point
moves along a level surface, the value of
remains fixed.
Contents [PD.1] [PD.2] [PD.3] [PD.4] [PD.5] [PD.6] [PD.7] [PD.8]
This document created by Scientific WorkPlace 4.1.