Chapter 5: Multivariable Functions

So far, we have focused on functions of a single variable, where each input is a single number $x$ and each output is a single number $y=f(x)$. Many situations, however, involve relationships between more than one independent variable.

When working with two independent variables, say $x$ and $y$, it is natural to consider ordered pairs $(x,y)$, where each coordinate is a real number. The set of all such pairs is denoted by ${\mathbb{R}}^2$ and is often thought of as the Cartesian plane. Similarly, ordered triples $(x,y,z)$ form ${\mathbb{R}}^3$, which we interpret as three-dimensional space. More generally, ${\mathbb{R}}^n$ denotes the set of all ordered $n$-tuples $(x_1,x_2,\dots ,x_n)$, where each coordinate is a real number.

Visualizing Functions of Multiple Variables

When $n=1$, the graph of a function $f(x)$ can be drawn in a two-dimensional coordinate system. When $n=2$, we can represent the graph in three dimensions, with the third axis showing the value of $f(x,y)$. For $n\ge 3$, it is no longer possible to directly visualize the graph in physical space, but other techniques, such as level curves and function traces, can be used to represent the function’s behavior.

To illustrate these ideas, we extend the crop yield model we studied earlier, in Chapter 3, to include additional factors.

Example: Extended Crop Yield Model

In reality, crop yield depends on more than just fertilizer amount. Another important factor is rainfall, denoted by $y$ (in millimeters over a growing season of about $120$ days, i.e., $\approx 4$ months). We now model crop yield as a function of two variables:

$$f(x,y)=\text{crop yield for fertilizer }x\text{ and rainfall }y$$

Here, $f:{\mathbb{R}}^2\to \mathbb{R}$ assigns a real-valued yield to each ordered pair $(x,y)$ within a suitable domain (e.g., $x>0$ kg/ha of fertilizer and $y>0$ mm of rainfall).

Because $f$ depends on two variables, its graph lives in three dimensions: the horizontal plane represents $x$ (fertilizer) and $y$ (rainfall), while the vertical axis represents $f(x,y)$ (yield). Although 3D graphs are possible, they can be difficult to interpret—especially for decision-making—so we often use level curves and function traces instead.

Level Curves or Contours

Level curves show where the function has the same value, making it easier to identify trade-offs and regions of interest.

In the crop yield model, a level curve for $f$ represents all combinations of fertilizer and rainfall that yield the same harvest.

For a fixed yield $z_0$, the level curve is:

$$\{(x,y)\mid f(x,y)=z_0\}$$

For example, the level curve for $f(x,y)=4.4\text{t/ha}$ shows all fertilizer-rainfall combinations producing a yield of $4.4$ tonnes per hectare.

From a contour plot, we can answer questions such as:

• "If I want $4.4$ t/ha, how can I trade fertilizer for rainfall?"
• "Where is the optimal combination of fertilizer and rainfall for maximum yield?"

Level curves are especially useful for visualizing decision boundaries and trade-offs when multiple factors influence an outcome.

Function Traces

Function traces help us examine cross-sections of the surface by fixing one variable and varying the other.

For example:

• Fix fertilizer at $150$ kg/ha and vary rainfall: The trace shows how yield changes with rainfall for that fertilizer level.

  

• Fix rainfall at $450$ mm and vary fertilizer: The trace shows how yield changes with fertilizer for that rainfall level.

  

From these traces, we can identify tipping points, such as the fertilizer amount beyond which adding more no longer increases yield.

Computing Level Curves and Traces

The task of finding level curves and traces reduces to solving equations. The algebraic and graphical techniques are the same as those used for curves in two dimensions, but here they are applied to cross-sections and slices of surfaces.