cal8 - Chapter Eight f : Rn R 8.1 Introduction We shall now...

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8.1 Chapter Eight f R R n : 8.1 Introduction We shall now turn our attention to the very important special case of functions that are real, or scalar, valued. These are sometimes called scalar fields . In the very, but important, special subcase in which the dimension of the domain space is 2, we can actually look at the graph of a function. Specifically, suppose f : R R n . The collection S x x x f x x x = = {( , , ) : ( , ) } 1 2 3 1 2 3 R 3 is called the graph of f . If f is a reasonably nice function, then S is what we call a surface. We shall see more of this later. Let us now return to the general case of a function f : R R n . The derivative of f is just a row vector f f x f x f x n '( ) x = 1 2 L . It is frequently called the gradient of f and denoted grad f or f 8.2 The Directional Derivative In the applications of scalar fields it is of interest to talk of the rate of change of the function in a specified direction. Suppose, for instance, the function T x y z ( , , ) gives the temperature at points ( , , ) x y z in space, and we might want to know the rate at which the temperature changes as we move in a specified direction. Let f : R R n , let a R n , and let u R n be a vector such that | | u = 1 Then the directional derivative of f at a in the direction of the vector u is defined to be D f d dt f t t u a a u ( ) ( ) = + = 0 .
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8.2 Now that we are experts on the Chain Rule, we know at once how to compute such a thing. It is simply D f d dt f t f t u a a u u ( ) ( ) = + = ∇ ⋅ = 0 . Example The surface of a mountain is the graph of f x y x y ( , ) = - - 700 5 2 2 . In other words, at the point ( x , y ), the height is f ( x , y ). The positive y -axis points North, and, of course, then the positive x -axis points East. You are on the mountain side above the point (2, 4) and begin to walk Southeast. What is the slope of the path at the starting point? Are you going uphill or downhill? (Which!?). The answers to these questions call for the directional derivative. We know we are at the point a = ( , ) 2 4 , but we need a unit vector u in the direction we are walking. This is, of course, just u = - 1 2 1 1 ( , ) . Next we compute the gradient = - - f x y x y ( , ) [ , ] 2 10 . At the point a this becomes = - - f ( , ) [ , ] 2 4 2 40 , and at last we have ∇ ⋅ = - + = f u 2 40 2 38 2 . This gives us the slope of the path; it is positive so we are going uphill. Can you tell in which direction the path will be level? Another Example The temperature in space is given by T x y z x y yz ( , , ) = + 2 3 . From the point (1,1,1), in which direction does the temperature increase most rapidly? We clearly need the direction in which the directional derivative is largest. The directional derivative is simply =∇ T T u | |cos θ , where θ is the angle between T and u . Anyone can see that this will be largest when θ = 0. Thus T in creases most rapidly in
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8.3 the direction of the gradient of T . Here that direction is [ , , ] 2 3 2 3 2 xy x z yz + . At (1,1,1), this becomes [2, 2, 3].
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cal8 - Chapter Eight f : Rn R 8.1 Introduction We shall now...

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