Chapter 16: Gradients; Extreme Values; Differentials

Home , Extreme value theorem, Mean value theorem

Section 16.1 Differentiability and Gradient Section 16.5 Local Extreme Values a. Definition: Differentiability a. Definition b. Definition: Gradient b. Theorem 16.5.2 c. Theorem 16.1.3 c. Critical Points, Stationary Points, Saddle Points d. Differentiability Implies Continuity d. Second-Partials Test

Section 16.2 Gradients and Directional Derivatives Section 16.6 Absolute Extreme Values a. Elementary Formulas a. Definition: Absolute Extreme Values b. Directional Derivative b. Definition: Bounded Sets c. Theorem 16.2.4 c. Extreme-Value Theorem d. Theorem 16.2.5 d. Two Variables: Absolute Extreme Values e. Rate of Change Section 16.7 Maxima and Minima with Side Conditions Section 16.3 The Mean-Value Theorem: The Chain Rule a. Lagrange Multiplier a. The Mean-Value Theorem b. Illustration b. Connected Sets c. Theorem 16.3.2 Section 16.8 Differentials d. Theorem 16.3.3 a. Illustration of a Differential e. The Chain Rule b. Differentials and Increments f. Another Formulation of the Chain Rule c. Illustration g. Implicit Differentiation Section 16.9 Reconstructing a Function from its Gradient Section 16.4 The Gradient as a Normal; Tangent Lines and a. Reconstruction: Parts 1, 2 and 3 Tangent Planes b. Simply Connected a. Gradient Vectors c. Theorem 16.9.2 b. Normal and Tangent Vectors c. Tangent and Normal Lines d. Functions of Three Variables e. Tangent Planes f. Equations for the Normal Line g. Surface z = g(x,y) and the Normal Line

Example For the function f (x, y) = x2 + y2.

f(xh+−) f( x) = fx( + hy12,, + h) − fxy( ) 22 =xh + ++ yh − x22 + y ( 12) ( )  = + ++22 [22xh1 yh 2]  h 12 h 2 =[22xyi + jh] ⋅+ h

The remainder g(h) = ||h||2 is o(h): 2 h as h → 0, =h0 → thus ∇=∇=+f(x) fxyxy( ,) 22 ij h

Differentiability Implies Continuity As in the one-variable case, differentiability implies continuity:

To see this, write f(xh+) − ff( x) =∇( xh) ⋅+ o( h) and note that f(xh+−=∇⋅+≤∇⋅+) f( x) f( xh) of( h) ( xh) o( h)

As h → 0, ∇ff(xh) ⋅ ≤∇( x) h →0 and o(h) →0

It follows that f (x + h) − f (x) → 0 and therefore f (x + h) → f (x).

Some Elementary Formulas In many respects gradients behave just as derivatives do in the one-variable case. In particular, if ∇ f (x) and ∇ g(x) exist, then ∇ [ f (x) + g(x)], ∇[αf (x)], and ∇ [ f (x)g(x)] all exist, and

Since the directional derivative gives the rate of change of the function in that direction, it is clear that

A nonempty open set U (in the plane or in three-space) is said to be connected if any two points of U can be joined by a polygonal path that lies entirely in U. You can see such a set pictured in Figure 16.3.1. The set shown in Figure 16.3.2 is the union of two disjoint open sets. The set is open but not connected: it is impossible to join a and b by a polygonal path that lies within the set.

Another Formulation of Theorem 16.3.4 The chain rule for functions of one variable, gives

In the two-variable case, the z-term drops out and we have

Implicit Differentiation

Consider now a curve in the xy-plane C : f (x, y) = c. We can view C as the c-level curve of f and conclude from (16.4.1) that the gradient

is perpendicular to C at (x0, y0). We call it a normal vector. The vector

is perpendicular to the gradient: ∂∂ff ∂∂ ff ∇⋅=fxy( ,,) txy( ) ( xy ,,) ( xy) −( xy ,,0) ( xy) = 00 00∂∂xy 00 00 ∂∂ yx 00 00 It is therefore a tangent vector.

Tangent Line

Normal Line

Functions of Three Variables

A point x lies on the tangent plane through x0 iff

This is an equation for the tangent plane in vector notation. In Cartesian coordinates the equation takes the form

The normal line to the surface f (x, y, z) = c at a point x0 = (x0, y0, z0) on the surface is the line which passes through (x0, y0, z0) parallel to ∇ f (x0). Thus, ∇ f (x0) is a direction vector for the normal line and

is a vector equation for the line. In scalar parametric form, equations for the normal line can be written

A surface of the form z = g(x, y) can be written in the form f (x, y, z) = 0 by setting f (x, y, z) = g(x, y) − z. If g is differentiable, so is f .

If ∇ g(x0, y0) = 0, then both partials of g are zero at (x0, y0) and the equation reduces to

In this case the tangent plane is horizontal. Scalar parametric equations for the line normal to the surface z = g(x, y) at the point (x0, y0, z0) can be written

In the one-variable case we know that if f has local extreme value at x0, then

f´(x0) = 0 or f´(x0) does not exist.

We have a similar result for functions of several variables.

Salas, Hille, Etgen Calculus: One and Several Variables Main Menu Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Local Extreme Values Two Variables We suppose for the moment that f = f (x, y) is defined on an open connected set and is continuously differentiable there. The graph of f is a surface z = f (x, y). Where f has a local maximum, the surface has a local high point. Where f has a local minimum, the surface has a local low point. Where f has either a local maximum or a local minimum, the gradient is 0 and therefore the tangent plane is horizontal.

Critical points at which the gradient is zero are called stationary points. The stationary points that do not give rise to local extreme values are called saddle points. Salas, Hille, Etgen Calculus: One and Several Variables Main Menu Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Local Extreme Values

Two Variables The procedure we use for finding the absolute extreme values can be outlined as follows:

1. Determine the critical points. These are the interior points at which the gradient is zero (the stationary points) and the interior points at which the gradient does not exist.

2. Determine the points on the boundary that can possibly give rise to extreme values. At this stage this is a one-variable process.

3. Evaluate f at the points found in Steps 1 and 2.

4. The greatest of the numbers found in Step 3 is the absolute maximum; the least is the absolute minimum.

The Method of Lagrange Let f be a function of two or three variables which is continuously differentiable on some open set U. We take C : r = r (t), t ∈ I to be a curve that lies entirely in U and has at each point a nonzero tangent vector r'(t).

Such a scalar λ has come to be called a Lagrange multiplier.

We begin by reviewing the one-variable case. If f is differentiable at x, then for small h, the increment Δf = f (x + h) − f (x) can be approximated by the differential d f = f´(x) h. For a geometric view of Δf and df, see Figure 16.8.1. We write ∆≅f df

is called the increment of f , and the dot product

is called the differential (more formally, the total differential). As in the one-variable case, for small h, the differential and the increment are approximately equal:

Part 1 Show how to find f (x, y) given its gradient

∂∂ff ∇=f( xy,,) ( xy) ij +( xy ,) ∂∂xy

Part 2 Show that, although all gradients ∇ f (x, y) are expressions of the form

P(x, y) i + Q(x, y) j

(set P = ∂ f/∂x and Q = ∂ f/∂y), not all such expressions are gradients.

Part 3 Recognize which expressions P(x, y) i + Q(x, y) j are actually gradients.