Section 7.1: Examples of Functions of Several Variables
A function need not be expressed in terms of a formula.
Traditionally, the variables of a function of several variables are written as
or if there are more than three variables, using
subscripts
.
The functions themselves are usually written using symbols such a
and if it is important to list the variables as, eg
,
, or
.
For function of two variables, one may graph this function by plotting the
solutions to
in three space.
Examples:
A contour map is precisely such a graph. Here the variables are the lattitude and longitude of a point on the Earth and the function gives the altitude.
(More on contour plots)
Suppose that one wishes to produce a structure in the shape of
a rectangular box. The material for the floor costs
per square foot, the material for the walls costs
per
square foot, and the material for the roof costs
per square foot.
Write the total cost as a function of
, the width,
, the
length, and
, the height, of the structure.
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Area of roof | ![]() |
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Total cost | ![]() |
Floor cost ![]() ![]() |
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Section 7.2: Partial Derivatives
If is a function of several variables, then for any
fixed value of
and
, say,
and
, the function
is a function of the single variable
.
As such, it makes sense to compute the derivative of , or what is the
same thing, the derivative of
with respect to
:
For a function of one variable, , the derivative of
at
is defined as a limit:
For a function of several variables, partial derivatives are defined by the same kind of limit.
In general, to compute the partial derivative of a function with respect to some variable, treat the function as a function of that single variable with all the other named variables regarded as constants.
If is a constant and
a natural number, then the formula
Instead, we could consider this monomial as a function of three variables
(at least for
) and the above formula
expresses
Let
. Compute
,
, and
.
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Let
. Compute
and
.
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Unlike a curve, a surface has many tangent lines at each point. The partial derivatives give the slopes of the tangent lines at a point in a specific direction.
More precisely, the partial derivative at a point
of a function
with respect
to
is the slope of the tangent line to the graph
of
at
along the direction where all coordinates
save
are held fixed.
As with derivatives of a function of a single variable, partial
derivatives may be interpreted as rates of change. In this case,
is the rate at which
changes
relative to changes in the
-variable with all other variables held
fixed.
Let
. Compute and interpret
and
.
. So the slope of the tangent line in the
direction at
is
.
. That is, the slope of the
tangent line in the
-direction is
.
Notice that is increasing in the
-direction while it is
decreasing in the
-direction.
A partial derivative of a function is itself a function and may be differentiated again.
It is a non-trivial, though true, theorem that for a sufficiently
smooth function the order of differentiation is immaterial. That is,
Let
. Compute
,
and
.
and
.
So,
and
, while
(or
we may compute
).