MAIN CONCEPTS
The direction from  in which  has its maximum rate of change is in the direciton of  and this maximum rate of change is Get the Printer Friendly Version

Suppose that  is defined for all  in some sphere about a point . As we approach point  from different vectors, we find that  is increasing in some directions, decreasing or remaing constant. To find the vector that gives us the maximum or minimum rate of change of , we simply use two simple theorems.

Let  and its first partial derivatives be continuous in some sphere about  and assume .

First theorem: The direction from  in which  has its maximum rate of change is in the direciton of  and this maximum rate of change is .
Second theorem: The direction from  in which  has its minimum rate of change is in the direciton of  and this minimum rate of cahnge is .

The two theorems are essentially opposites of each other and are not too hard to prove with the help of the cosine function. We shall show the proof here.

Let  be any unit vector drawn as an arrow from . Then,

Knowing that we can choose from different unit vectors, we want to choose  such that this directional derivative is as large as possible. We now define the angle  to be the angle between  and  as shown.

 

By using vector algebra, we write

since . Here comes the reasoning. We maximize  by maximizing  and in turn maximizing . This occurs when  or  meaning to say that  is in the same direction as . Moreover, this gives us  as our theorem says.

By the same argument,  has its lowest value at  or . In this case,  is in the same direciton as  and .

We use these two theorems to find normal vectors and tangent planes which are coming up next.