More on Curvature and Arc Length
Contour Problems

Let us begin with an example.

Example 1.

Consider the function f\left (x\right ) =x^{2} defined for every x in \left [ -1 ,1\right ]. Domain of this function is the set

(1)   \begin{equation*}\{x : -1 \leq x \leq 1\} \end{equation*}

and the range of f is all numbers in \left [0 ,1\right ] that is

(2)   \begin{equation*}\{y :y =x^{2} , -1 \leq x \leq 1\} \end{equation*}

This function can be shown as

We also know the familiar graph of this function

Example 2.

Now what if we are given a function f which depends on two variables, say x and y ? Consider the function z =f\left (x ,y\right ) =x^{2} +y^{2} defined on the unit disk. The domain of this function is the set

(3)   \begin{equation*}\{\left (x ,y\right ) :x^{2} +y^{2} \leq 1\} \end{equation*}

The range of f is the set of all possible outputs of f that is

(4)   \begin{equation*}\{z :z =x^{2} +y^{2} ,x^{2} +y^{2} \leq 1\} \end{equation*}

This function can be shown as

The graph of f is shown below. Note that the origin is the lowest point on the graph of f.

The function z =f\left (x ,y\right ) in example 2 is said to be a single-valued function of two variables, single output and a pair of inputs. Let us look at another example.

Example 3.

Define z =f\left (x ,y\right ) =\sqrt{x^{2} +y^{2}} defined on the first quadrant of the unit disk. The domain of this function is the set

(5)   \begin{equation*}\{\left (x ,y\right ) :x^{2} +y^{2} \leq 1 ,x \geq 0 ,y \geq 0\} \end{equation*}

The range of f is

(6)   \begin{equation*}\{z :z =\sqrt{x^{2} +y^{2}} ,x^{2} +y^{2} \leq 1 ,x \geq 0 ,y \geq 0\} \end{equation*}

This function can be shown as

And the graph in 3-space is

Remark.

One can extend what we have done in previous examples to define functions of more than two variables. Such functions are called functions of several variables. In this case, it will not be possible to graph them but we still can get a sense of how they behave. To begin, consider the following example.

Example 4.

Consider the function z =f(x ,y) =x^{2} +y^{2} in example 2. If we fix z =\frac{1}{2}, then \frac{1}{2} =x^{2} +y^{2} is a horizontal cut in the graph of f at z =\frac{1}{2}. We like to see the shadow of such cut in xy-plane. In fact, for any fixed z in \left [0 ,1\right ] we get a cut of the graph, say z =0 ,z =1 or z =\frac{1}{4} . The shadows of these cuts are shown below. If one cuts the graph of the function at z =2 , of course nothing is obtained since z =2 is not in the range of f .

Such shadow curves are called level curves or in general contours.

Example 5.

Consider the function z =f\left (x ,y\right ) =x +y . We are interested in seeing the contours for f for z =0 ,1 ,5. Start by graphing each of

(7)   \begin{equation*}0 =x +y , 1 =x +y, 5 =x +y\end{equation*}

Example 6.

We draw the contours associated to z =f\left (x ,y\right ) =xy when z =\frac{1}{2} ,1 ,2

The contours of a function of three variables is called level surfaces. This is due to the fact that for w =f\left (x ,y ,z\right ) and w =c , for some c , the contour f\left (x ,y ,z\right ) =c is a surface in 3-space.

Example 7.

We are given w =f\left (x ,y ,z\right ) =x^{2} +y^{2} +z^{2} defined for any point in 3-space. Each level surface of f is a sphere, centered at the origin. For example, given w =4 , we obtain a level surface of f which is x^{2} +y^{2} +z^{2} =4 that is a sphere of radius 2.

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