Multivariate Calculus - Bruce E. Shapiro

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80 LECTURE 11. GRADIENTS AND THE DIRECTIONAL DERIVATIVE Hence At the point (1,1,1) we have 200(xi + yj + zk) ∇T (x, y, z) = − (10 + x 2 + y 2 + z 2 ) 2 ∂ ∂x ∣ −200x ∣∣∣(1,1,1) 10 + x 2 + y 2 + z 2 = −200 169 and similarly for the other two partial derivatives. Hence the direction of steepest increase in temperature is ∇T (x, y, z)| (1,1,1) = − 200 (i + j + k) 169 Theorem 11.5 The gradient vector is perpendicular to the level curves of a function. To see why this theorem is true, recall the definition of a level curve: it is a curve along which the value of z = f(x, y) is a constant. Thus if you move along a tangent vector to a level curve, the function will not change, and the directional derivative is zero. Since the directional derivative is the dot product of the gradient and the direction of motion, which in this case is the tangent vector to the level curve, this dot product is zero. But a dot product of two non-zero vectors can only be zero if the two vectors are perpendicular to one another. Thus the gradient must be perpendicular to the level curve. Corollary 11.1 Let f(x, y, z) = 0 describe a surface in 3D space. Then the threedimensional gradient vector ∇f is a normal vector to the surface. This result follows because a surface f(x, y, z) = 0 is a level surface of a function w = f(x, y, z). The value of f(x, y, z) does not change. So if you consider any tangent vector, the directional derivative along the tangent vector is zero. Consequently the tangent vector is perpendicular to the gradient. Revised December 6, 2006. Math 250, Fall 2006
Lecture 12 The Chain Rule Recall the chain rule from <strong>Calculus</strong> I: To find the derivative of some function u = f(x) with respect to a new variable t, we calculate du dt = du dx dx dt We can think of this as a sequential process, as illustrated in figure 12.1. 1. Draw a labeled node for each variable. The original function should be the furthest to the left, and the desired final variable the node furthest to the right. 2. Draw arrows connecting the nodes. 3. Label each arrow with a derivative. The variable on the top of the derivative corresponds to the variable the arrow is coming from and the variable on the bottom of the derivative is the variable the arrow is going to. 4. follow the path described by the labeled arrows. The derivative of the variable at the start of the path with respect to the variable at the end of the path is the product of the derivatives you meet along the way. Figure 12.1: Visualization of the chain rule for a function of a single variable Now suppose u is a function of two variables x and y rather than just one. Then we need to have two arrows emanating from u, one to each variable. This is illustrated in figure 12.2. The procedure is modified as follows: 1. Draw a node for the original function u, each variable it depends on x, y, ..., and the final variable that we want to find the derivative with respect to t. 81
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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.
Page 41 and 42: Lecture 4 Lines and Curves in 3D We
Page 43 and 44: LECTURE 4. LINES AND CURVES IN 3D 3
Page 49 and 50: Lecture 5 Velocity, Acceleration, a
Page 51 and 52: LECTURE 5. VELOCITY, ACCELERATION,
Page 57 and 58: Lecture 6 Surfaces in 3D The text f
Page 59 and 60: Lecture 7 Cylindrical and Spherical
Page 61 and 62: 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
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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 and 114: Lecture 14 Unconstrained Optimizati
Page 115 and 116: 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
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LECTURE 16. DOUBLE INTEGRALS OVER R
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LECTURE 16. DOUBLE INTEGRALS OVER R
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Lecture 17 Double Integrals over Ge
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LECTURE 17. DOUBLE INTEGRALS OVER G
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Lecture 18 Double Integrals in Pola
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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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