Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences

Throughout the text thus far we have focused investigating functions of the form \(y=f(x)\text{,}\) with one independent and one dependent variable. Such functions can be represented in two dimensions, using two numerical axes that allow us to identify every point in the plane with two numbers. We now shift our focus to three-dimensional space; to identify every point in three dimensions we require three numerical values. The obvious way to make this association is to add one new axis, perpendicular to the \(x\)- and \(y\)-axes we already understand. We could, for example, add a third axis, the \(z\)-axis, with the positive \(z\)-axis coming straight out of the page, and the negative \(z\)-axis going out the back of the page. This is difficult to work with on a printed page, so more often we draw a view of the three axes from an angle:

You must then imagine that the \(z\)-axis is perpendicular to the other two. Just as we have investigated functions of the form \(y=f(x)\) in two dimensions, we will investigate three dimensions largely by considering functions; now the functions will (typically) have the form \(z=f(x,y)\text{.}\) Due to the fact that we are used to having the result of a function graphed in the vertical direction, it is somewhat easier to maintain that convention in three dimensions. To accomplish this, we normally rotate the axes so that \(z\) points up; investigate by rotating the axes below:

It is now fairly simple to understand some “shapes” in three dimensions that correspond to simple conditions on the coordinates. In two dimensions the equation \(x=1\) describes the vertical line through \((1,0)\text{.}\) In three dimensions, it still describes all points with \(x\)-coordinate 1, but this is now a plane, as in Figure 6.2.

Recall the very useful distance formula in two dimensions which comes directly from the Pythagorean Theorem: the distance between points \((x_1,y_1)\) and \((x_2,y_2)\) is \(\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\text{.}\) What is the distance between two points \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\) in three dimensions? Geometrically, we want the length of the long diagonal labelled \(c\) in the “box” in Figure 6.3. Since \(a\text{,}\) \(b\text{,}\) \(c\) form a right triangle, \(a^2+b^2=c^2\text{.}\) \(b\) is the vertical distance between \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\text{,}\) so \(b=|z_1-z_2|\text{.}\) The length \(a\) runs parallel to the \(x\)-\(y\)-plane, so it is simply the distance between \((x_1,y_1)\) and \((x_2,y_2)\text{,}\) that is, \(a^2=(x_1-x_2)^2+(y_1-y_2)^2\text{.}\) Now we see that \(c^2=(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2\) and \(c=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}\text{.}\)

It is sometimes useful to give names to points, for example we might let \(P_1=(x_1,y_1,z_1)\text{,}\) or more concisely we might refer to the point \(P_1(x_1,y_1,z_1)\text{,}\) and subsequently use just \(P_1\text{.}\) Distance between two points in either two or three dimensions is sometimes denoted by \(d\text{,}\) so for example the formula for the distance between \(P_1(x_1,y_1,z_1)\) and \(P_2(x_2,y_2,z_2)\) might be expressed as

In two dimensions, the distance formula immediately gives us the equation of a circle: the circle of radius \(r\) and center at \((h,k)\) consists of all points \((x,y)\) at distance \(r\) from \((h,k)\text{,}\) so the equation is \(r=\sqrt{(x-h)^2+(y-k)^2}\) or \(r^2=(x-h)^2+(y-k)^2\text{.}\) Now we can get the similar equation \(r^2=(x-h)^2+(y-k)^2+(z-l)^2\text{,}\) which describes all points \((x,y,z)\) at distance \(r\) from \((h,k,l)\text{,}\) namely, the sphere with radius \(r\) and center \((h,k,l)\text{.}\)