Multivariate Calculus - Bruce E. Shapiro

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102 LECTURE 14. UNCONSTRAINED OPTIMIZATION 1. All points on the boundary of J; 2. All stationary points, namely, any point where ∇f(x, y) = 0; 3. All singular points, namely, any points where f(x, y) or ∇f(x, y) are undefined. The candidate extrema are sometimes called critical points; some authors restrict the use of this term to describe the stationary points. Stationary points occur when ∇f(x, y) = 0; this requires all of the derivatives to be zero: ∂f ∂x = 0 = ∂f ∂y Recall from one-variable calculus that three different things could happen at a stationary point where f ′ (a) = 0: • f(x) could have a local maximum at x = a, as, for example, occurs for f(x) = −(x − a) 2 • f(x) could have a local minimum at x = a, as, for example, occurs for f(x) = (x − a) 2 • f(x) could have an inflection point at x = a, as, for example, occurs for f(x) = (x − a) 3 For multivariate functions there are three types of stationary points: local maxima, local minima, and saddle points. Saddle points are the generalization of inflection points. At a saddle point, in some vertical cross-sections, the function appears to have a local minimum, while in other vertical cross-sections, the function appears to have a local maximum. Example 14.1 Find the stationary points of the function f(x, y) = x 2 − 7xy + 12y 2 − y Solution. The function has stationary points when f x (x, y) = f y (x, y) = 0. Differentiating, ∂f ∂x = 0 ⇒ ∂ ∂x (x2 − 7xy + 12y 2 − y) = 2x − 7y = 0 ⇒ y = 2x/7 ∂f ∂y = 0 ⇒ ∂ ∂y (x2 − 7xy + 12y 2 − y) = −7x + 24y − 1 = 0 ⇒ 24y = 7x + 1 Combining the two results, 48 49x + 7 x = 7x + 1 = ⇒ x = −7 7 7 Revised December 6, 2006. Math 250, Fall 2006
LECTURE 14. UNCONSTRAINED OPTIMIZATION 103 Figure 14.1: Top: Topology of a solid around a saddle point; the canonical function is z = y 2 − x 2 with a saddle point at the origin, although many other functions display saddle points. Bottom: Origin of the nomenclature. The draft saddle model 1583 is sold by Midwest Leather Co. for $955 [from http://www.draftsource.com/Midwest Leather]. Math 250, Fall 2006 Revised December 6, 2006.
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Multivariate Calculus in 25 Easy Le
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Contents 1 Cartesian Coordinates 1
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Preface: A note to the Student Thes
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CONTENTS v The order in which the m
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Examples of Typical Symbols Used Sy
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CONTENTS ix Table 1: Symbols Used i
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Lecture 1 Cartesian Coordinates We
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LECTURE 1. CARTESIAN COORDINATES 3
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Lecture 2 Vectors in 3D Properties
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LECTURE 2. VECTORS IN 3D 11 Figure
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LECTURE 2. VECTORS IN 3D 13 Definit
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LECTURE 2. VECTORS IN 3D 15 Hence u
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LECTURE 2. VECTORS IN 3D 17 and the
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LECTURE 2. VECTORS IN 3D 19 The Equ
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Lecture 3 The Cross Product Definit
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LECTURE 3. THE CROSS PRODUCT 23 Pro
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LECTURE 3. THE CROSS PRODUCT 25 Exa
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LECTURE 3. THE CROSS PRODUCT 27 5.
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Lecture 4 Lines and Curves in 3D We
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LECTURE 4. LINES AND CURVES IN 3D 3
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Lecture 5 Velocity, Acceleration, a
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LECTURE 5. VELOCITY, ACCELERATION,
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Lecture 6 Surfaces in 3D The text f
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Lecture 7 Cylindrical and Spherical
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LECTURE 7. CYLINDRICAL AND SPHERICA
Page 63 and 64: Lecture 8 Functions of Two Variable
Page 65 and 66: LECTURE 8. FUNCTIONS OF TWO VARIABL
Page 71 and 72: Lecture 9 The Partial Derivative De
Page 73 and 74: LECTURE 9. THE PARTIAL DERIVATIVE 6
Page 79 and 80: Lecture 10 Limits and Continuity In
Page 81 and 82: LECTURE 10. LIMITS AND CONTINUITY 6
Page 83 and 84: LECTURE 10. LIMITS AND CONTINUITY 7
Page 85 and 86: Lecture 11 Gradients and the Direct
Page 87 and 88: LECTURE 11. GRADIENTS AND THE DIREC
Page 93 and 94: Lecture 12 The Chain Rule Recall th
Page 95 and 96: LECTURE 12. THE CHAIN RULE 83 The p
Page 97 and 98: LECTURE 12. THE CHAIN RULE 85 Examp
Page 99 and 100: LECTURE 12. THE CHAIN RULE 87 becau
Page 101 and 102: LECTURE 12. THE CHAIN RULE 89 Solut
Page 103 and 104: LECTURE 12. THE CHAIN RULE 91 Examp
Page 105 and 106: Lecture 13 Tangent Planes Since the
Page 107 and 108: LECTURE 13. TANGENT PLANES 95 Solut
Page 109 and 110: LECTURE 13. TANGENT PLANES 97 Multi
Page 111 and 112: LECTURE 13. TANGENT PLANES 99 Accor
Page 113: Lecture 14 Unconstrained Optimizati
Page 117 and 118: LECTURE 14. UNCONSTRAINED OPTIMIZAT
Page 129 and 130: Lecture 15 Constrained Optimization
Page 131 and 132: LECTURE 15. CONSTRAINED OPTIMIZATIO
Page 139 and 140: Lecture 16 Double Integrals over Re
Page 141 and 142: LECTURE 16. DOUBLE INTEGRALS OVER R
Page 147 and 148: Lecture 17 Double Integrals over Ge
Page 149 and 150: LECTURE 17. DOUBLE INTEGRALS OVER G
Page 157 and 158: Lecture 18 Double Integrals in Pola
Page 159 and 160: LECTURE 18. DOUBLE INTEGRALS IN POL
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LECTURE 18. DOUBLE INTEGRALS IN POL
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Lecture 19 Surface Area with Double
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LECTURE 19. SURFACE AREA WITH DOUBL
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LECTURE 19. SURFACE AREA WITH DOUBL
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Lecture 20 Triple Integrals Triple
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LECTURE 20. TRIPLE INTEGRALS 163 Fi
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LECTURE 20. TRIPLE INTEGRALS 165 so
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LECTURE 20. TRIPLE INTEGRALS 167 Tr
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Lecture 21 Vector Fields Definition
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LECTURE 21. VECTOR FIELDS 171 Defin
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LECTURE 21. VECTOR FIELDS 173 Examp
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Lecture 22 Line Integrals Suppose t
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LECTURE 22. LINE INTEGRALS 177 Exam
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LECTURE 22. LINE INTEGRALS 179 ener
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LECTURE 22. LINE INTEGRALS 181 The
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LECTURE 22. LINE INTEGRALS 183 so t
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LECTURE 22. LINE INTEGRALS 185 wher
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Lecture 23 Green’s Theorem Theore
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LECTURE 23. GREEN’S THEOREM 193 T
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Lecture 24 Flux Integrals & Gauss
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LECTURE 24. FLUX INTEGRALS & GAUSS
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LECTURE 24. FLUX INTEGRALS & GAUSS
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Lecture 25 Stokes’ Theorem We hav
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LECTURE 25. STOKES’ THEOREM 203 S
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Multivariate Calculus - Bruce E. Shapiro

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