9.7. Gradient of a Scalar Field. Directional Derivatives. A. The Gradient. 1. Let f: R3 ! R be a scalar ¯eld, that is, a function of three variables. The gradient of f, denoted rf, is the vector ¯eld given by " # @f @f @f @f @f @f rf = ; ; = i + j + k: @x @y @y @x @y @z 2. Symbolically, we can consider r to be a vector of di®erential operators, that is, @ @ @ r = i + j + k: @x @y @z Then rf is symbolically the \vector" r \times" the \scalar" f. 3. Note that rf is a vector ¯eld so that at each point P , rf(P ) is a vector, not a scalar. B. Directional Derivative. 1. Recall that for an ordinary function f(t), the derivative f 0(t) represents the rate of change of f at t and also the slope of the tangent line at t. The gradient provides an analogous quantity for scalar ¯elds. 2. When considering the rate of change of a scalar ¯eld f(x; y; z), we talk about its rate of change in a direction b. (So that b is a unit vector.) 3. The directional derivative of f at P in direction b is f(P + tb) ¡ f(P ) Dbf(P ) = lim = rf(P ) ¢ b: t!0 t Note that in this de¯nition, b is assumed to be a unit vector. C. Maximum Rate of Change. 1. The directional derivative satis¯es Dbf(P ) = rf(P ) ¢ b = jbjjrf(P )j cos γ = jrf(P )j cos γ where γ is the angle between b and rf(P ). 2. Therefore, jDbf(P )j is largest therefore when γ = 0 or γ = ¼. If γ = 0 then Dbf(P ) is maximized and if γ = ¼, Dbf(P ) is minimized. Note also that the directional derivative is zero when γ = ¼=2, that is, in the direction perpendicular to rf(P ). 1 D. Normal Vector to a Surface. 1. Given a curve C in the plane, we are used to talking about the line tangent to C at a given point P 2 C. Analogously, given a surface S in 3-space, we can talk about the plane tangent to S at a point P 2 S. What is the equation for this plane? 2. If a plane is tangent to S at P , this means that the tangent line of every curve in S passing through P must lie in this plane. That is, if r(t) = [x(t); y(t); z(t)] is in S and 0 if r(t0) = P , then n ¢ r (t0) = 0 where n is a vector normal to the plane. 3. If S is given by the equation f(x; y; z) = 0 then this means that ¯ ¯ ¯ ¯ d ¯ d ¯ f(x(t); y(t); z(t))¯ = f(r(t))¯ = rf(r(t )) ¢ r0(t ) = rf(P ) ¢ r0(t ) = 0 dt ¯ dt ¯ 0 0 0 t=t0 t=t0 so that rf(P ) is normal to the tangent plant to S at P . E. Conservative Fields. 1. Given a vector ¯eld F, is it always true that there is a scalar ¯eld f such that F = rf? (This would be analogous to ¯nding an \antiderivative" for F.) The answer is no. 2. If there is such an f, we say that the vector ¯eld F is conservative. 2.