Multivariate Calculus - Bruce E. Shapiro

More documents

Recommendations

Info

$NAME: Math 103L: Exercise on Functions ... - Bruce E. Shapiro$

$Introducing Latex - Bruce E. Shapiro$

$Applied Differential Equations, Math 280 at CSUN - Bruce E. Shapiro$

$Math 103 Section 1.2: Linear Equations and ... - Bruce E. Shapiro$

170 LECTURE 21. VECTOR FIELDS x y v x v y -2 -2 1 -1/2 -2 -1 1/2 -1/2 -2 0 0 -1/2 -2 1 -1/2 -1/2 -2 2 -1 -1/2 -1 -2 1 -1/4 -1 -1 1/2 -1/4 -1 0 0 -1/4 -1 1 -1/2 -1/4 -1 2 -1 -1/4 0 -2 1 0 0 -1 1/2 0 0 0 0 0 0 1 -1/2 0 0 2 -1 0 1 -2 1 1/4 1 -1 1/2 1/4 1 0 0 1/4 1 1 -1/2 1/4 1 2 -1 1/4 2 -2 1 1/2 2 -1 1/2 1/2 2 0 0 1/2 2 1 -1/2 1/2 2 2 -1 1/2 We can then sketch the plot using the data in the table. Figure 21.1: A visualization of the vector field −iy/2 + jx/4. Revised December 6, 2006. Math 250, Fall 2006
LECTURE 21. VECTOR FIELDS 171 Definition 21.2 A scalar field is a function f(x, y, z) : D ⊂ R 3 ↦→ R that associates a scalar (number) with each point in a subset D of R 3 . Definition 21.3 The gradient of a scalar field f(x, y, z) is the vector field ( ∂f gradf(x, y, z) = ∇f(x, y, z) = ∂x , ∂f ∂y , ∂f ) ∂z Definition 21.4 The gradient operator is the vector operator ( ∂ grad = ∇ = ∂x , ∂ ∂y , ∂ ) ∂z (21.1) (21.2) The gradient operator is the 3-dimensional analogue of the differential operators such as ∂ ∂y or d dx Just as with these scalar operators, the gradient operator works by operating on anything written to the right of it. The operations are distributed through to the different components of the vector operator. Thus, for example, and ( ∂ ∂x , ∂ ∂y , ∂ ∂z ) f = ( ∂f ∂x , ∂f ∂y , ∂f ∂z ( ∂ ∂x , ∂ ∂y , ∂ ) ( ∂(uv) uv = ∂z ∂x , ∂(uv) ∂y , ∂(uv) ) ∂z Equation 21.3 can be used to derive the product rule for gradients: ) (21.3) ∇(uv) = u∇v + v∇u (21.4) Definition 21.5 A gradient field F(x, y, z) is a vector field that is the gradient of some scalar field f(x, y, z). If such a function f exists, it is called the potential function of the gradient field F and F is said to be a conservative vector field. Example 21.2 Find the gradient field F(x, y, z) corresponding to the function f(x, y, z) = x 2 z + 3xy. Solution. The gradient field is F(x, y, z) = ∇f(x, y, z) = ∇ ( x 2 z + 3xy ) ( ∂ = ∂x , ∂ ∂y , ∂ ) (x 2 z + 3xy ) ∂z ( ∂ ( = x 2 z + 3xy ) , ∂ ( x 2 z + 3xy ) , ∂ ( x 2 z + 3xy )) ∂x ∂y ∂z = ( 2xz + 3y, 3x, x 2) Math 250, Fall 2006 Revised December 6, 2006.
Page 1 and 2:
Multivariate Calculus in 25 Easy Le
Page 3 and 4:
Contents 1 Cartesian Coordinates 1
Page 5 and 6:
Preface: A note to the Student Thes
Page 7 and 8:
CONTENTS v The order in which the m
Page 9 and 10:
Examples of Typical Symbols Used Sy
Page 11 and 12:
CONTENTS ix Table 1: Symbols Used i
Page 13 and 14:
Lecture 1 Cartesian Coordinates We
Page 15 and 16:
LECTURE 1. CARTESIAN COORDINATES 3
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Lecture 2 Vectors in 3D Properties
Page 23 and 24:
LECTURE 2. VECTORS IN 3D 11 Figure
Page 25 and 26:
LECTURE 2. VECTORS IN 3D 13 Definit
Page 27 and 28:
LECTURE 2. VECTORS IN 3D 15 Hence u
Page 29 and 30:
LECTURE 2. VECTORS IN 3D 17 and the
Page 31 and 32:
LECTURE 2. VECTORS IN 3D 19 The Equ
Page 33 and 34:
Lecture 3 The Cross Product Definit
Page 35 and 36:
LECTURE 3. THE CROSS PRODUCT 23 Pro
Page 37 and 38:
LECTURE 3. THE CROSS PRODUCT 25 Exa
Page 39 and 40:
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 45 and 46:
Page 47 and 48:
Page 49 and 50:
Lecture 5 Velocity, Acceleration, a
Page 51 and 52:
LECTURE 5. VELOCITY, ACCELERATION,
Page 53 and 54:
Page 55 and 56:
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 67 and 68:
Page 69 and 70:
Page 71 and 72:
Lecture 9 The Partial Derivative De
Page 73 and 74:
LECTURE 9. THE PARTIAL DERIVATIVE 6
Page 75 and 76:
Page 77 and 78:
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 89 and 90:
Page 91 and 92:
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 and 114:
Lecture 14 Unconstrained Optimizati
Page 115 and 116:
LECTURE 14. UNCONSTRAINED OPTIMIZAT
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
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
Page 167 and 168: Lecture 19 Surface Area with Double
Page 169 and 170: LECTURE 19. SURFACE AREA WITH DOUBL
Page 171 and 172: LECTURE 19. SURFACE AREA WITH DOUBL
Page 173 and 174: Lecture 20 Triple Integrals Triple
Page 175 and 176: LECTURE 20. TRIPLE INTEGRALS 163 Fi
Page 177 and 178: LECTURE 20. TRIPLE INTEGRALS 165 so
Page 179 and 180: LECTURE 20. TRIPLE INTEGRALS 167 Tr
Page 181: Lecture 21 Vector Fields Definition
Page 185 and 186: LECTURE 21. VECTOR FIELDS 173 Examp
Page 187 and 188: Lecture 22 Line Integrals Suppose t
Page 189 and 190: LECTURE 22. LINE INTEGRALS 177 Exam
Page 191 and 192: LECTURE 22. LINE INTEGRALS 179 ener
Page 193 and 194: LECTURE 22. LINE INTEGRALS 181 The
Page 195 and 196: LECTURE 22. LINE INTEGRALS 183 so t
Page 197 and 198: LECTURE 22. LINE INTEGRALS 185 wher
Page 203 and 204: Lecture 23 Green’s Theorem Theore
Page 205 and 206: LECTURE 23. GREEN’S THEOREM 193 T
Page 207 and 208: Lecture 24 Flux Integrals & Gauss
Page 209 and 210: LECTURE 24. FLUX INTEGRALS & GAUSS
Page 211 and 212: LECTURE 24. FLUX INTEGRALS & GAUSS
Page 213 and 214: Lecture 25 Stokes’ Theorem We hav
Page 215 and 216: LECTURE 25. STOKES’ THEOREM 203 S
show all

Multivariate Calculus - Bruce E. Shapiro

Create successful ePaper yourself

Delete template?

Save as template?