Here we review the concept of the gradient and the directional derivative and prove the formula for the directional derivative discussed earlier.
Let f : U → ℝ where U is an open subset of ℝ^{n} and suppose f is differentiable on U. Thus if x ∈ U,
 (12.16) 
Now we can prove the formula for the directional derivative in terms of the gradient.
Proof:


Now lim_{t→0}

as claimed. ■
Example 12.9.2 Let f

Note this vector which is given is already a unit vector. Therefore, from the above, it is only necessary to find ∇f

Therefore, ∇f

Because of (12.17) it is easy to find the largest possible directional derivative and the smallest possible directional derivative. That which follows is a more algebraic treatment of an earlier result with the trigonometry removed.
Proposition 12.9.3 Let f : U → ℝ be a differentiable function and let x ∈ U. Then
 (12.18) 
and
 (12.19) 
Furthermore, the maximum in (12.18) occurs when v = ∇f
Proof: From (12.17) and the Cauchy Schwarz inequality,

and so for any choice of v with

The proposition is proved by noting that if v = −∇f
For a different approach to the proposition, see Problem 7 which follows.
The conclusion of the above proposition is important in many physical models. For example, consider some material which is at various temperatures depending on location. Because it has cool places and hot places, it is expected that the heat will flow from the hot places to the cool places. Consider a small surface having a unit normal n. Thus n is a normal to this surface and has unit length. If it is desired to find the rate in calories per second at which heat crosses this little surface in the direction of n it is defined as J ⋅ nA where A is the area of the surface and J is called the heat flux. It is reasonable to suppose the rate at which heat flows across this surface will be largest when n is in the direction of greatest rate of decrease of the temperature. In other words, heat flows most readily in the direction which involves the maximum rate of decrease in temperature. This expectation will be realized by taking J = −K∇u where K is a positive scalar function which can depend on a variety of things. The above relation between the heat flux and ∇u is usually called the Fourier heat conduction law and the constant K is known as the coefficient of thermal conductivity. It is a material property, different for iron than for aluminum. In most applications, K is considered to be a constant but this is wrong. Experiments show that this scalar should depend on temperature. Nevertheless, things get very difficult if this dependence is allowed. The constant can depend on position in the material or even on time.
An identical relationship is usually postulated for the flow of a diffusing species. In this problem, something like a pollutant diffuses. It may be an insecticide in ground water for example. Like heat, it tries to move from areas of high concentration toward areas of low concentration. In this case J = −K∇c where c is the concentration of the diffusing species. When applied to diffusion, this relationship is known as Fick’s law. Mathematically, it is indistinguishable from the problem of heat flow.
Note the importance of the gradient in formulating these models.
+ class=”left” align=”middle”(v)12.10. THE GRADIENT AND TANGENT PLANES