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.
Area of the floor | |||
Area of walls | |||
Area of roof |
Total cost | Floor cost Wall cost Roof cost | ||
Floor area Wall area Roof area | |||
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 .
Let . Compute and .
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 ).