Modeling Functions of Multiple Variables: Topics of Partial Derivatives

Introduction to Functions of Multiple Variables

Prior to now, we have quite frequently worked with functions that relate a single variable to another variable. These functions tend to take the form:

f(x)=x

Here, f(x) represents the traditional variable y. However, in this article, we investigate higher functions that rely on relationships between multiple variable inputs. We call these functions of multiple variables. Consider an example in the form of temperature. We can investigate the manner in which temperature (T) varies in accordance with latitude (x) and longitude (y). In this case, we model the multi-variable function as:

T=f(x,y)

Before embarking on this subject, it should be noted that the foundations of this topic requires minor conceptualization of vector calculus, and some linear algebra as well. Consider checking out these topics before moving forward:

Vector Calculus:

  1. General Vector Calculus
  2. Dot Product and Cross Product 
  3. Vector Function
  4. Vector Differentiation and Integration
  5. Arc Length and Arc Curvature

Linear Algebra:

  1. Gaussian Method of Solving Linear Systems
  2. Describing Solutions of Linear Systems

Defining Functions of Two Variables

Functions of two variables generally adhere to the definition as follows:

\text{A function ‘f’ of two variables establishes a rule which relates} \newline \text{variables as an ordered pair (x,y) in a set domain D of real} \newline \text{numbers. If ‘D’ is the domain and ‘f’ is the range resulting}\newline \text{from the values ‘x’ and ‘y’, then we may say:} \newline \{f(x,y)\enspace | \enspace (x,y) \in D\}

In understanding this definition, we can actually say that z=f(x,y). In this manner, the value of the function in the z-dimension depends both upon the values in the x and y dimension. This is a consequence of the relationship defined by ‘f’. By this relationship, we can say that the variables ‘x’ and ‘y’ are independent variables while the variable ‘z’ is the dependent variable.

Example of Two-Variable Function

With our definition for the two-variable function in mind, let us put this to use. Let’s evaluate the following function at f(3,2):

f(x,y)=\frac{\sqrt{x+y+1}}{x-1}\newline f(3,2)=\frac{\sqrt{3+2+1}}{3-1}\newline f(3,2)=\frac{\sqrt{6}}{2}

One aspect we ought note is that while these functions have their domain in the real numbers ‘R’, the function is not defined for all variables. We might consider the circumstance wherein x=1, which consequentially render a denominator of zero. So, it is sometimes of use to define the domain of the function ‘f’. We may do so as follows:

D=\{(x,y)\enspace |\enspace x+y+1 \geq 0, \enspace x \not=1

Graphs of Two-Variable Functions

In addition to feasible computation, graphical representation of two-variable functions is quite operable. When we work with a function of the form x=f(x,y), then ‘f’ can be readily be plotted from the set of all points (x,y,z) in three dimensional space. While in the case of a single-variable function produces a curve of the form y=f(x), the two-variable function produces a surface. We conceptualize this via the intercepts of the curve. Consider the following function:

f(x,y)=6-3x-2y

Just by looking at the equation, we can observe that the z-intercept is (0,0,6). Setting z=0, we can identify the x and y-intercepts as (2,0,0) and (0,3,0),respectively. If we were to graph this, we would observe that this function produces a surface. It appears as follows:

We consider this function a linear function, which are those that take the form:

f(x,y)=ax+by+c

Linear functions comprised of multiple variables constitute a plane. However, there are certainly much more complex graphs of these multi-variable functions. However, it is often preferential to model these using computer programs. Consider the function of the parabola, which often takes the form:

f(x)=x^2

Now, consider the function of the ellipse, which takes the form:

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

Suppose that we have a function that combines the relationships defined by these functions. Consider this function:

f(x,y)=4x^2+y^2

This function is an elliptic paraboloid. As such, in three dimensions, it exhibits behaviors both of a parabola and an ellipse. Take a look at its computer generated graph.

If you look near the top of the structure, the elliptic shape is quite obvious. This results from the elliptic function involved in our multi-variable equation. However, if you take a look at the lines on the surface, you will notice the parabolic nature of the equation as well. Evidently, the graphs of these multi-variable functions can take on some complex features.

Modeling Multi-Variable Functions With Level Curves

Aside from Cartesian graphs, another methodology by which to model multi-variable equations is through level curves. In level curves, the density of lines together is indicative of their relative heights in three dimensions. These level curves, however, consist of two dimensions.

Level curves are defined by functions ‘f’ comprised of two variables such that f(x,y)=k. In this case, ‘k’ is a constant. In such a form, for every pair of points (x,y), values for ‘k’ which are explicitly labeled. This value of ‘k’ is demonstrative of the value ‘z’ in a three dimensional space.

Let us put this knowledge into practice with an example. Consider the following function:

f(x,y)=4x^2+y^2+1

Let us attempt to model this multi-variable function using level curves. We might note that this is quite similar to the elliptic parabloid from the previous example, except for its translation one unit in the positive z-dimension. So, the first thing we do is set the function equal to ‘k’:

k=4x^2+y^2+1

Now, if we substitute a value in for ‘k’, then it may be obvious that there are many combinations of ‘x’ and’y’ which can exist at that particular value. If we examine a series of values for k, we may note that the level curves will appear as follows:

In this image, the elliptic nature of the function is quite obvious. However, if we pay attention to the density of the lines, we will see that in the y-dimension the density increases. This may not be immediately intuitive, but the reason is obvious when we graph the function in three dimensions.

Here, the structure of the elliptic parboloid is brought to light. As a result, we can see the utility in using level curves in modeling multi-variable functions. This utility, primarily, is the simplified graphing of functions in three dimensions using only two dimensions. All it takes is a little mental gymnastics to make the leap to comprehending its three dimensional constitution.

Functions of Three or More Variables

Though we are limited to the three dimensional modeling of function in space, we are still capable of working with higher ordered functions. This is particularly true in the case of functions that may have, for example, three variables. Functions of three variables consist of a functional assignment to an ordered pair of the form (x,y,z). These functions take the form f(x,y,z). For example, in our temperature example, we said temperature depends on latitutde and longitude such that T=f(x,y). However, we can increase the complexity of this system, saying that temperature depends on latitude, longitude, and time such that T=f(x,y,t).

We are not limited to the use of only three variables, however. In fact, any number of variables can be considered in a function. A function of ‘n’ variables establishes a rule that assigns a number z=f(x1,x2, … ,xn) to the n-tuple (x1,x2, … ,xn) of real numbers. We may then say that:

C=f(x_1,x_2,…,x_n)=c_1x_1+c_2x_2+ \cdot \cdot \cdot+c_nx_n

Functions of Multiple Variables

The present article discussed at length the intricacies of working with functions of multiple variables, beginning by demonstrating the form of these functions. We then illuminated the methodical models which can be used to represent these functions; in particular, graphing and level curves. If any of this has been insufficient, an article will soon be produced tying all of these concepts together in the form of an example. This example will be an elaboration of the Cobb and Douglas function. Understanding these concepts will be critical to comprehending the use and utility of partial derivatives, so do not be overly concerned with ‘spending too much time’ on this. More on this subject found here.

Consider checking out these topics as well:

Vector Calculus:

  1. General Vector Calculus
  2. Dot Product and Cross Product 
  3. Vector Function
  4. Vector Differentiation and Integration
  5. Arc Length and Arc Curvature

Linear Algebra:

  1. Gaussian Method of Solving Linear Systems
  2. Describing Solutions of Linear Systems

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