22.9. THE GRADIENT AND TANGENT PLANES 483

Proof: From 22.16 and the Cauchy Schwarz inequality, |Dv f (x)| ≤ |∇ f (x)| and so forany choice of v with |v|= 1,−|∇ f (x)| ≤ Dv f (x)≤ |∇ f (x)|. The proposition is provedby noting that if v =−∇ f (x)/ |∇ f (x)|, then

Dv f (x) = ∇ f (x) · (−∇ f (x)/ |∇ f (x)|) =−|∇ f (x)|2 / |∇ f (x)|=−|∇ f (x)|

while if v = ∇ f (x)/ |∇ f (x)|, then

Dv f (x) = ∇ f (x) · (∇ f (x)/ |∇ f (x)|) = |∇ f (x)|2 / |∇ f (x)|= |∇ f (x)| .

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 thehot places to the cool places. Consider a small surface having a unit normal n. Thus n isa normal to this surface and has unit length. If it is desired to find the rate in calories persecond at which heat crosses this little surface in the direction of n it is defined as J ·nAwhere A is the area of the surface and J is called the heat flux. It is reasonable to supposethe rate at which heat flows across this surface will be largest when n is in the direction ofgreatest rate of decrease of the temperature. In other words, heat flows most readily in thedirection which involves the maximum rate of decrease in temperature. This expectationwill be realized by taking J = −K∇u where K is a positive scalar function which candepend on a variety of things. The above relation between the heat flux and ∇u is usuallycalled the Fourier heat conduction law and the constant K is known as the coefficient ofthermal conductivity. It is a material property, different for iron than for aluminum. Inmost applications, K is considered to be a constant but this is wrong. Experiments showthat this scalar should depend on temperature. Nevertheless, things get very difficult if thisdependence is allowed. The constant can depend on position in the material or even ontime.

An identical relationship is usually postulated for the flow of a diffusing species. In thisproblem, something like a pollutant diffuses. It may be an insecticide in ground water forexample. Like heat, it tries to move from areas of high concentration toward areas of lowconcentration. 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 isindistinguishable from the problem of heat flow.

Note the importance of the gradient in formulating these models.

22.9 The Gradient and Tangent PlanesLet S ≡ {x ∈ Rp : g(x) = 0} be a level surface. We assume ∇g(y) ̸= 0 for some y ∈ S.Then a tangent plane at y will be of the form {x ∈ Rp : n·(x−y) = 0} . The problem isto find n which is a vector which is perpendicular to every vector from y to x and wewant this to be a real tangent plane. The way you can achieve this is to require that nbe perpendicular to the direction vector of every smooth curve through y which lies in S.One such n is obtained from ∇g(y) . Indeed, if t → x(t) is a curve through y such thatx(0) = y, then g(x(t)) = 0 and so from the chain rule, ∇g(y) ·x′ (0) = 0. Thus a suitablechoice for n will be ∇g(y). Of course, this is a specious argument without the implicitfunction theorem which gives existence of such smooth curves in the level surface. See thechapter on this theorem presented later.