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

We still have not answered one of our first questions about the steepness of a surface: starting at a point on a surface given by \(f(x,y)\text{,}\) and walking in a particular direction, how steep is the surface? We are now ready to answer the question.

We already know roughly what has to be done: as shown in Figure 7.3, we extend a line in the \(x\)-\(y\)-plane to a vertical plane, and we then compute the slope of the curve that is the cross-section of the surface in that plane. The major stumbling block is that what appears in this plane to be the horizontal axis, namely the line in the \(x\)-\(y\)-plane, is not an actual axis—we know nothing about the “units” along the axis. Our goal is to make this line into a \(t\) axis; then we need formulas to write \(x\) and \(y\) in terms of this new variable \(t\text{;}\) then we can write \(z\) in terms of \(t\) since we know \(z\) in terms of \(x\) and \(y\text{;}\) and finally we can simply take the derivative.

So we need to somehow “mark off” units on the line, and we need a convenient way to refer to the line in calculations. It turns out that we can accomplish both by using the vector form of a line. Suppose that \(\vect{u}\) is a unit vector \(\langle u_1,u_2\rangle\) in the direction of interest. A vector equation for the line through \((x_0,y_0)\) in this direction is \(\vect{v}(t)=\langle u_1t+x_0,u_2t+y_0\rangle\text{.}\) The height of the surface above the point \((u_1t+x_0,u_2t+y_0)\) is \(g(t)=f(u_1t+x_0,u_2t+y_0)\text{.}\) Because \(\vect{u}\) is a unit vector, the value of \(t\) is precisely the distance along the line from \((x_0,y_0)\) to \((u_1t+x_0,u_2t+y_0)\text{;}\) this means that the line is effectively a \(t\) axis, with origin at the point \((x_0,y_0)\text{,}\) so the slope we seek is

Here we have used the Chain Rule and the derivatives \({d\over dt}(u_1t+x_0)=u_1\) and \({d\over dt}(u_2t+y_0)=u_2\text{.}\) The vector \(\langle f_x,f_y\rangle\) is very useful, so it has its own symbol, \(\nabla f\text{,}\) pronounced “del f” ; it is also called the gradient of \(f\text{.}\)

The slope of a surface given by \(z=f(x,y)\) in the direction of a (two-dimensional) vector \(\vect{u}\) is called the directional derivative of \(f\text{,}\) written \(D_{\vect{u}}f\text{.}\) The directional derivative immediately provides us with some additional information. We know that

if \(\vect{u}\) is a unit vector; \(\theta\) is the angle between \(\nabla f\) and \(\vect{u}\text{.}\) This tells us immediately that the largest value of \(D_{\vect{u}}f\) occurs when \(\cos\theta=1\text{,}\) namely, when \(\theta=0\text{,}\) so \(\nabla f\) is parallel to \(\vect{u}\text{.}\) In other words, the gradient \(\nabla f\) points in the direction of steepest ascent of the surface, and \(|\nabla f|\) is the slope in that direction. Likewise, the smallest value of \(D_{\vect{u}}f\) occurs when \(\cos\theta=-1\text{,}\) namely, when \(\theta=\pi\text{,}\) so \(\nabla f\) is anti-parallel to \(\vect{u}\text{.}\) In other words, \(-\nabla f\) points in the direction of steepest descent of the surface, and \(-|\nabla f|\) is the slope in that direction.

If \(\nabla f\) is perpendicular to \(\vect{u}\text{,}\) \(D_{\vect{u}}f=|\nabla f|\cos(\pi/2)=0\text{,}\) since \(\cos(\pi/2)=0\text{.}\) This means that in either of the two directions perpendicular to \(\nabla f\text{,}\) the slope of the surface is 0; this implies that a vector in either of these directions is tangent to the level curve at that point. Starting with \(\nabla f=\langle f_x,f_y\rangle\text{,}\) it is easy to find a vector perpendicular to it: either \(\langle f_y,-f_x\rangle\) or \(\langle -f_y,f_x\rangle\) will work.

If \(f(x,y,z)\) is a function of three variables, all the calculations proceed in essentially the same way. The rate at which \(f\) changes in a particular direction is \(\nabla f\cdot\vect{u}\text{,}\) where now \(\nabla f=\langle f_x,f_y,f_z\rangle\) and \(\vect{u}=\langle u_1,u_2,u_3\rangle\) is a unit vector. Again \(\nabla f\) points in the direction of maximum rate of increase, \(-\nabla f\) points in the direction of maximum rate of decrease, and any vector perpendicular to \(\nabla f\) is tangent to the level surface \(f(x,y,z)=k\) at the point in question. Of course there are no longer just two such vectors; the vectors perpendicular to \(\nabla f\) describe the tangent plane to the level surface, or in other words \(\nabla f\) is a normal to the tangent plane.