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Chapter 16: Gradients; Extreme Values; Differentials

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Section 16.1 Differentiability and Gradient

a. Definition: Differentiability

b. Definition: Gradient

c. Theorem 16.1.3

d. Differentiability Implies Continuity

Section 16.2 Gradients and Directional Derivatives

a. Elementary Formulas

b. Directional Derivative

c. Theorem 16.2.4

d. Theorem 16.2.5

e. Rate of Change

Section 16.3 The Mean-Value Theorem: The Chain Rule

a. The Mean-Value Theorem

b. Connected Sets

c. Theorem 16.3.2

d. Theorem 16.3.3

e. The Chain Rule

f. Another Formulation of the Chain Rule

g. Implicit Differentiation

Section 16.4 The Gradient as a Normal; Tangent Lines and Tangent Planes

a. Gradient Vectors

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

Chapter 16: Gradients; Extreme Values; Differentials

Section 16.5 Local Extreme Values

a. Definition

b. Theorem 16.5.2

c. Critical Points, Stationary Points, Saddle Points

d. Second-Partials Test

Section 16.6 Absolute Extreme Values

a. Definition: Absolute Extreme Values

b. Definition: Bounded Sets

c. Extreme-Value Theorem

d. Two Variables: Absolute Extreme Values

Section 16.7 Maxima and Minima with Side Conditions

a. Lagrange Multiplier

b. Illustration

Section 16.8 Differentials

a. Illustration of a Differential

b. Differentials and Increments

c. Illustration

Section 16.9 Reconstructing a Function from its Gradient

a. Reconstruction: Parts 1, 2 and 3

b. Simply Connected

c. Theorem 16.9.2

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentiability and Gradient

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentiability and Gradient

Example

For the function f (x, y) = x 2 + y 2 .

The remainder g(h) = ||h|| 2 is o(h):

as h → 0,

( ) ( ) ( ) ( )

( ) ( )

[ ]

[ ]

1 2

2 2 2 2

1 2

2 2

1 2 1 2

2

, ,

2 2

2 2

f f f x h y h f x y

x h y h x y

xh yh h h

x y

+ − = + + −

   

=  + + +  −  + 

 

= + +  + 

= + ⋅ +

x h x

i j h h

2

= →

h h 0

h thus f ( ) x = ∇ f x y ( ) , = 2 x i + 2 y j

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentiability and Gradient

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentiability and Gradient

Differentiability Implies Continuity

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

To see this, write

( ) ( ) ( ) ( )

f x + hf x = ∇ f x h ⋅ + o h and note that

( ) ( ) ( ) ( ) ( ) ( )

f x + hf x = ∇ f x h ⋅ + o h ≤ ∇ f x h ⋅ + o h As h → 0,

It follows that

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

( ) ( ) 0 and ( ) 0

f f o

x h ⋅ ≤ ∇ x hh

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

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

Gradients and Directional Derivatives

∇ ∇ ∇

∇ ∇

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Gradients and Directional Derivatives

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Gradients and Directional Derivatives

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Gradients and Directional Derivatives

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Gradients and Directional Derivatives

Since the directional derivative gives the rate of change of the function in

that direction, it is clear that

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Mean-Value Theorem: The Chain Rule

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Mean-Value Theorem: The Chain Rule

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.

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The Mean-Value Theorem: The Chain Rule

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The Mean-Value Theorem: The Chain Rule

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Mean-Value Theorem: The Chain Rule

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Mean-Value Theorem: The Chain Rule

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

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Mean-Value Theorem: The Chain Rule

Implicit Differentiation

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The Gradient as a Normal

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Gradient as a Normal

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 (x 0 , y 0 ). We call it a normal vector. The vector

is perpendicular to the gradient:

It is therefore a tangent vector.

( 0 , 0 ) ( 0 , 0 ) f ( 0 , 0 ) ( f 0 , 0 ) f ( 0 , 0 ) ( f 0 , 0 ) 0

f x y t x y x y x y x y x y

x y y x

∂ ∂ ∂ ∂

∇ ⋅ = − =

∂ ∂ ∂ ∂

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Gradient as a Normal

Tangent Line

Normal Line

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Gradient as a Normal

Functions of Three Variables

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Gradient as a Normal

A point x lies on the tangent plane through x 0 iff

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

coordinates the equation takes the form

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The normal line to the surface f (x, y, z) = c at a point x 0 = (x 0 , y 0 , z 0 ) on the surface is the line which passes through (x 0 , y 0 , z 0 ) parallel to f (x 0 ). Thus, f (x 0 ) is a direction vector for the normal line and

The Gradient as a Normal

∇ ∇

is a vector equation for the line. In scalar parametric form, equations for the

normal line can be written

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

The Gradient as a Normal

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(x 0 , y 0 ) = 0, then both partials of g are zero at (x 0 , y 0 ) 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 (x 0 , y 0 , z 0 ) can be written

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Local Extreme Values

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Local Extreme Values

In the one-variable case we know that if f has local extreme value at x 0 , then f ´ (x 0 ) = 0 or f ´ (x 0 ) does not exist.

We have a similar result for functions of several variables.

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Salas, Hille, Etgen Calculus: One and Several Variables 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.

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Local Extreme Values

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Absolute Extreme Values

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Absolute Extreme Values

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Absolute Extreme Values

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Absolute 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.

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Maxima and Minima with Side Conditions

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.

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Maxima and Minima with Side Conditions

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentials

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

∆ ≅

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentials

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:

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Differentials

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Reconstructing a Function from its Gradient

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

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

(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.

( ) , f ( ) , f ( ) ,

f x y x y x y

x y

∂ ∂

∇ = +

ij

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

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Reconstructing a Function from its Gradient

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Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

Reconstructing a Function from its Gradient

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