Mathematics
functions of
several variables
functions of
several variables
Partial derivatives
Differential
Linear approximation
Error calculation
Extrema of a function
© The scientific sentence. 2010
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Calculus I:
functions of several variables
Graphical representation of
a function of
two variables
Graphical representation of
a function of two variables
Before giving the definition of the graph of a function
of two variables, let's recall what is the graph of
a function of a variable.
• Definition :
f: D → R
x → f (x)
The graph Cf of f (function of a single variable) is
the set of points of the plane
of coordinates (x; f (x)) with x  D.
This is noted:
Cf = {(x, y) R2
| y = f (x), x D}
So to draw the graph of a function of one variable we
have added a new variable y. The graph is then a curve
in the plane R2.
For the functions of two variables x and y we will also add
a variable z and the graph will then be a surface of the space
R3.
• Definition
f: D → R
(x, y) → f (x, y)
The graph Sf of f (function of two variables) is the
set of points of space of coordinates (x; y; f (x, y))
with (x, y) D.
This is noted:
Sf = {(x, y, z) R3 |
z = f (x, y), (x, y) D}
Note :
Sf is a surface in R3.
At each point (x, y) D
corresponds to a point on the surface Sf. Here's how to
place the points in a frame.
To get familiar with the graphs of the functions of two variables here
some examples drawn by Gnuplot:
The Gnuplot code is:
reset
set xrange [-5:5]
set yrange [-5:5]
set zrange [-1:1]
set ticslevel 1
set xlabel "X"
set ylabel "Y"
set zlabel "Z"
set grid
set isosamples 40, 40
splot sin(x+y)
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