13.1 The Geometry of Real-Valued Functions

We launch our investigation of real-valued functions by developing methods for visualizing them. In particular, we introduce the notions of a graph, a level curve, and a level surface of such functions.

Functions and Mappings

Let \(f\) be a function whose domain is a subset \(A\) of \({\mathbb R}^n\) and with a range contained in \({\mathbb R}^m\). By this we mean that to each \({\bf x}=(x_1,\ldots,x_n)\in A,f\) assigns a value \(f({\bf x})\), an \(m\)-tuple in \({\mathbb R}^m\). Such functions \(f\) are called vector-valued functionsfootnote # if \(m\,{>}\,1\), and scalar-valued functions if \(m=1\). For example, the scalar-valued function \(f(x,y,z)=(x^2+y^2+z^2)^{-3/2}\) maps the set \(A\) of \((x,y,z)\neq (0,0,0)\) in \({\mathbb R}^3\) (\(n=3\), in this case) to \({\mathbb R}\) \((m=1)\). To denote \(f\) we sometimes write \[ f\colon\, (x,y,z) \mapsto (x^2+y^2+z^2)^{-3/2}. \]

Note that in \({\mathbb R}^3\) we often use the notation \((x,y,z)\) instead of \((x_1,x_2,x_3)\). In general, the notation \({\bf x}\mapsto f({\bf x})\) is useful for indicating the value to which a point \({\bf x}\in {\mathbb R}^n\) is sent. We write \(f\colon A\subset {\mathbb R}^n \rightarrow {\mathbb R}^m\) to signify that \(A\) is the domain of \(f\)(a subset of \({\mathbb R}^n\)) and the range is contained in \({\mathbb R}^m\). We also use the expression \(f\) maps \(A\) into \({\mathbb R}^m\). Such functions \(f\) are called functions of several variables if \(A\subset {\mathbb R}^n, n > 1\).

As another example we can take the vector-valued function \(g\colon\, {\mathbb R}^6 \rightarrow {\mathbb R}^2\) defined by the rule \[ g({\bf x}) = g(x_1,x_2,x_3,x_4,x_5,x_6) = \Big(x_1x_2x_3x_4x_5x_6,{\textstyle\sqrt{x_1^2 + x_6^2}}\Big). \]

The first coordinate of the value of \(g\) at \({\bf x}\) is the product of the coordinates of \({\bf x}\).

Functions from \({\mathbb R}^n\) to \({\mathbb R}^m\) are not just mathematical abstractions, they arise naturally in problems studied in all the sciences. For example, to specify the temperature \(T\) in a region \(A\) of space requires a function \(T \colon\, A\subset {\mathbb R}^3 \rightarrow {\mathbb R}\) \((n=3,m=1);\) thus, \(T(x,y,z)\) is the temperature at the point \((x,y,z)\). To specify the velocity of a fluid moving in space requires a map \({\bf V}\colon\, {\mathbb R}^4\rightarrow {\mathbb R}^3\), where \({\bf V}(x,y,z,t)\) is the velocity vector of the fluid at the point \((x,y,z)\) in space at time \(t\) (see Figure 13.1). Such a function \({\bf V}\) is called a time dependent vector field. To specify the reaction rate of a solution consisting of six reacting chemicals \(A,B, C,D,E,F\) in proportions \(x,y,z,w,u,v\) requires a map \(\sigma \colon\, U\subset {\mathbb R}^6\rightarrow {\mathbb R}\), where \(\sigma(x,y,z,w,u,v)\) gives the rate when the chemicals are in the indicated proportions.

Figure 13.1: A fluid in motion defines a vector field V by specifying the velocity of the fluid particles at each point in space and time.

To specify the cardiac vector (the vector giving the magnitude and direction of electric current flow in the heart) at time \(t\) requires a map \({\bf c}\colon\, {\mathbb R}\rightarrow {\mathbb R}^3, t \mapsto {\bf c} (t)\).

When \(f\colon\, U\subset {\mathbb R}^n\rightarrow {\mathbb R}\), we say that \(f\) is a real-valued function of n variables with domain U. The reason we say “\(n\) variables” is simply that we regard the coordinates of a point \({\bf x}=(x_1,\ldots,x_n)\in U\) as \(n\) variables, and \(f({\bf x})=f(x_1,\ldots,x_n)\) depends on these variables. We say “real-valued” because \(f(x_1,\ldots,x_n)\) is a real number. A good deal of our work will be with real-valued functions, so we give them special attention.

Graphs of Functions

For \(f\colon\, U\subset {\mathbb R}\rightarrow {\mathbb R}\) \((n=1)\), the graph of \(f\) is the subset of \({\mathbb R}^2\) consisting of all points \((x,f(x))\) in the plane, for \(x\) in \(U\). This subset can be thought of as a curve in \({\mathbb R}^2\). In symbols, we write this as \[ \hbox{ graph} f = \{ (x, f(x)) \in {\mathbb R}^2 \mid x \in U\}, \] where the curly braces mean “the set of all” and the vertical bar is read “such that.” Drawing the graph of a function of one variable is a useful device to help visualize how the function actually behaves (see Figure 13.2). It will be helpful to generalize the idea of a graph to functions of several variables. This leads to the following definition:

For the case \(n=1\), the graph is a curve in \({\mathbb R}^2\), while for \(n=2\), it is a surface in \({\mathbb R}^3\) (see Figure 13.2).

Figure 13.2: The graphs of (a) a function of one variable, and (b) a function of two variables.

Figure 13.3: Graph of \(f(x,y)=2x^2+5y^2\)

Figure 13.4: Different views of the graph of \( \displaystyle z= e^{-x^2-y^2}-e^{-(x-1)^2-(y-1)^2}\)

For \(n=3\), it is difficult to visualize the graph, because, since we are humans living in a three-dimensional world, it is hard for us to envisage sets in \({\mathbb R}^4\). To help overcome this handicap, we introduce the idea of a level set.

Level Sets, Curves, and Surfaces

Suppose \(f(x,y,z)= x^2 + y^2 + z^2\). A level set is a subset of \({\mathbb R}^3\) on which \(f\) is constant; for instance, the set where \(x^2+y^2+z^2= 1\) is a level set for \(f\). This we can visualize: It is just a sphere of radius 1 in \({\mathbb R}^3\). Formally, a level set is the set of \((x,y,z)\) such that \(f(x,y,z)=c\), where \(c\) is a constant. The behavior or structure of a function is determined in part by the shape of its level sets; consequently, understanding these sets aids us in understanding the function in question. Level sets are also useful for understanding functions of two variables \(f(x,y)\), in which case we speak of level curves or level contours.

The idea is similar to that used to prepare contour maps, where one draws lines to represent constant altitudes; walking along such a line would mean walking on a level path. In the case of a hill rising from the \(xy\) plane, a graph of all the level curves gives us a good idea of the function \(h(x,y)\), which represents the height of the hill at point \((x,y)\) (see Figure 13.5).

Figure 13.5: Level contours of a function are defined in the same manner as contour lines for a topographical map.

example 2

The function \(f\colon\, {\mathbb R}^2\rightarrow {\mathbb R},\) defined by \(f(x,y)=x+y+2\), has as its graph the inclined plane \(z=x+y+2\). This plane intersects the \(xy\) plane \((z=0)\) in the line \(y=-x-2\) and the \(z\) axis at the point \((0,0,2)\). For any value \(c \in {\mathbb R}\), the level curve of value \(c\) is the straight line \(y= - x+(c-2)\); or in symbols, the set \[ L_c = \{(x,y) \mid y = - x+ (c-2)\} \subset {\mathbb R}^2. \]

We indicate a few of the level curves of the function in Figure 13.6. This is a contour map of the function \(f\).

Figure 13.6: The level curves of \(f\hbox{(}x, y\hbox{)} = x + y + 2\) show the sets on which \(f\) takes a given value.

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Figure 13.7: The relationship of level curves of Figure 13.6 to the graph of the function \(f\hbox{(}x, y\hbox{)} = x + y + 2\), which is the plane \(z = x + y + 2\).

From level curves labeled with the value or “height” of the function, the shape of the graph may be inferred by mentally elevating each level curve to the appropriate height, without stretching, tilting, or sliding it. If this procedure is visualized for all level curves, \(L_c\)—that is, for all values \(c \in {\mathbb R}\), they will assemble to give the entire graph of \(f\), as indicated by the shaded plane in Figure 13.7. If the graph is visualized using a finite number of level curves, a contour model is produced. If \(f\) is a smooth function, its graph will be a smooth surface, and so the contour model, mentally smoothed over, gives a good impression of the graph.

EXAMPLE 3 Elliptic Paraboloid

Sketch the contour map of \(f(x,y)= x^2+3y^2\) and comment on the spacing of the contour curves.

Solution The level curves have equation \(f(x,y)=c\), or \[ x^2+3y^2=c \]

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• For \(c > 0\), the level curve is an ellipse.
• For \(c=0\), the level curve is just the point \((0,0)\) because \(x^2+3y^2=0\) only for \((x,y)=(0,0)\).
• The level curve is empty if \(c<0\) because \(f(x,y)\) is never negative.

The graph of \(f(x,y)\) is an elliptic paraboloid (Figure 13.8). As we move away from the origin, \(f(x,y)\) increases more rapidly. The graph gets steeper, and the level curves get closer together.

Figure 13.8: \(f(x,y) = x^2 + 3y^2\). Contour interval \(m=10\).

EXAMPLE 4 Hyperbolic Paraboloid

Sketch the contour map of \(g(x,y) = x^2-3y^2\).

Solution The level curves have equation \(g(x,y)=c\), or \[ x^2-3y^2=c \]

• For \(c \neq 0\), the level curve is the hyperbola \( x^2-3y^2=c\).
• For \(c=0\), the level curve consists of the two lines \(x=\pm \sqrt 3y\) because the equation \(g(x,y) = 0\) factors as follows: \[ x^2-3y^2=0 = (x-\sqrt 3y)(x+\sqrt 3 y) = 0 \]

The graph of \(g(x,y)\) is a hyperbolic paraboloid (Figure 13.9). When you stand at the origin, \(g(x,y)\) increases as you move along the \(x\)-axis in either direction and decreases as you move along the \(y\)-axis in either direction. Furthermore, the graph gets steeper as you move out from the origin, so the level curves get closer together.

Figure 13.9: \(g(x,y) = x^2 - 3y^2\). Contour interval \(m=10\).

For \(n\)-dimensional spaces we have the following definition:

The Method of Sections

By a section of the graph of \(f\) we mean the intersection of the graph and a (vertical) plane.

example 5

Use the methods of contours and sections to describe the graph of the quadratic function \[ f\colon \, {\mathbb R}^2\rightarrow {\mathbb R},(x,y) \mapsto x^2+y^2. \]

solution The graph is the paraboloid of revolution \(z=x^2+y^2\), oriented upward from the origin, around the \(z\) axis. The level curve of value \(c\) is empty for \(c<0\); for \(c>0\) the level curve of value \(c\) is the set \(\{(x,y) \mid x^2 + y^2 = c\}\), a circle of radius \(\sqrt{c}\) centered at the origin. Thus, raised to height \(c\) above the \(xy\) plane, the level set is a circle of radius \(\sqrt{c}\), indicating a parabolic shape (see Figure 13.10 and Figure 13.11).

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Figure 13.10: Some level curves for the function \(f\hbox{(}x, y\hbox{)} = x^2 + y^2\).

Figure 13.11: Level curves in Figure 13.10 raised to the graph.

Vertical sections of \(f\) are parabolas: For example, if \(P_1\) is the \(xz\) plane in \({\mathbb R}^3\), defined by \(y=0\), then the section of \(f\) in Example 5 is the set \[ S_1=P_1 \cap \hbox{ graph } f = \{ (x,y,z) \mid y = 0,z=x^2 \}, \] which is a parabola in the \(xz\) plane. Similarly, if \(P_2\) denotes the \(yz\) plane, defined by \(x=0\), then the section \[ S_2=P_2 \cap \hbox{ graph } f = \{(x,y,z)\mid x=0, z=y^2 \} \] is a parabola in the \(yz\) plane (see Figure 13.12). It is usually helpful to compute at least one section to complement the information given by the level sets.

Figure 13.12: Two sections of the graph of \(f\hbox{(}x, y\hbox{)} = x^2 + y^{\,2}\).

We have seen how the methods of sections and level sets can be used to understand the behavior of a function and its graph; these techniques can be quite useful to people who desire comprehensive visualization of complicated data. There are many computer programs available to do this, and we show the results of one such program in Figure 13.13.

Figure 13.13: Computer-generated graph of \(z = \hbox{(}x^2 + 3y^{\,2}\hbox{)}\, \exp\, \hbox{(}1-x^2 - y^{\,2}\hbox{)}\) represented in three ways: (a) by sections, (b) by level curves on a graph, and (c) by level curves in the \(xy\) plane.

Here are some more examples of level surfaces:

Figure 13.14: Level surfaces of \(g(x,y,z) = x^2+y^2-z^2\).