Michael Corral: Vector Calculus

More documents

Recommendations

Info

186 CHAPTER 4. LINE AND SURFACE INTEGRALS ☛ ✟ ✡Exercises ✠ A For Exercises 1-6, find the Laplacian of the function f(x,y,z) in Cartesian coordinates. 1. f(x,y,z)= x+y+z 2. f(x,y,z)= x 5 3. f(x,y,z)=(x 2 +y 2 +z 2 ) 3/2 4. f(x,y,z)=e x+y+z 5. f(x,y,z)= x 3 +y 3 +z 3 6. f(x,y,z)=e −x2 −y 2 −z 2 7. Find the Laplacian of the function in Exercise 3 in spherical coordinates. 8. Find the Laplacian of the function in Exercise 6 in spherical coordinates. 9. Let f(x,y,z)= z x 2 in Cartesian coordinates. Find∇f in cylindrical coordinates. +y2 10. For f(r,θ,z)=re r +z sinθe θ +rze z in cylindrical coordinates, find div f and curl f. 11. For f(ρ,θ,φ)=e ρ +ρ cosθe θ +ρe φ in spherical coordinates, find div f and curl f. B For Exercises 12-23, prove the given formula (r =‖r‖ is the length of the position vector field r(x,y,z)= xi+yj+zk). 12.∇(1/r)=−r/r 3 13.∆(1/r)=0 14.∇·(r/r 3 )=0 15.∇(lnr)=r/r 2 16. div(F+G)=div F+div G 17. curl(F+G)=curl F + curl G 18. div(f F)= f div F+F·∇f 19. div(F×G)=G·curl F − F·curl G 20. div(∇f×∇g)=0 21. curl(f F)= f curl F + (∇f )×F 22. curl(curl F)=∇(div F) −∆F 23.∆(fg)= f∆g + g∆ f + 2(∇f·∇g) C 24. Prove Theorem 4.17. 25. Derive the gradient formula in cylindrical coordinates:∇F= ∂F ∂r e r+ 1 r 26. Use f=u∇v in the Divergence Theorem to prove: ∂F ∂θ e θ+ ∂F ∂z e z (a) Green’s first identity: S (u∆v + (∇u)·(∇v))dV= Σ (u∇v)·dσ (b) Green’s second identity: S (u∆v − v∆u)dV= Σ (u∇v−v∇u)·dσ 27. Suppose that∆u=0(i.e. u is harmonic) over 3 . Define the normal derivative ∂u ∂n of u over a closed surfaceΣwith outward unit normal vector n by ∂u ∂n = D nu=n·∇u. Show that Σ ∂u ∂n dσ=0. (Hint: Use Green’s second identity.)
Bibliography Abbott, E.A., Flatland, 7th edition. New York: Dover Publications, Inc., 1952 Classic tale about a creature living in a 2-dimensional world who encounters a higherdimensional creature, with lots of humor thrown in. Anton, H. and C. Rorres, Elementary Linear Algebra: Applications Version, 8th edition. New York: John Wiley & Sons, 2000 Standard treatment of elementary linear algebra. Bazaraa, M.S., H.D. Sherali and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition. New York: John Wiley & Sons, 1993 Thorough treatment of nonlinear optimization. Farin, G., Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, 2nd edition. San Diego, CA: Academic Press, 1990 An intermediate-level book on curve and surface design. Hecht, E., Optics, 2nd edition. Reading, MA: Addison-Wesley Publishing Co., 1987 An intermediate-level book on optics, covering a wide range of topics. Hoel, P.G., S.C. Port and C.J. Stone, Introduction to Probability Theory, Boston, MA: Houghton Mifflin Co., 1971 An excellent introduction to elementary, calculus-based probability theory. Lots of good exercises. Jackson, J.D., Classical Electrodynamics, 2nd edition. New York: John Wiley & Sons, 1975 An advanced book on electromagnetism, famous for being intimidating. Most of the mathematics will be understandable after reading the present book. Marion, J.B., Classical Dynamics of Particles and Systems, 2nd edition. New York: Academic Press, 1970 Standard intermediate-level treatment of classical mechanics. Very thorough. O’Neill, B., Elementary Differential Geometry, New York: Academic Press, 1966 Intermediate-level book on differential geometry, with a modern approach based on differential forms. 187
Page 1:
1 0.8 0.6 0.4 z 0.2 0 -0.2 -0.4 -10
Page 4 and 5:
About the author: Michael Corral is
Page 6 and 7:
iv Preface matter, anyone can make
Page 8 and 9:
vi Contents 4.4 Surface Integrals a
Page 10 and 11:
2 CHAPTER 1. VECTORS IN EUCLIDEAN S
Page 12 and 13:
Page 14 and 15:
Page 16 and 17:
Page 18 and 19:
10 CHAPTER 1. VECTORS IN EUCLIDEAN
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
66 CHAPTER 2. FUNCTIONS OF SEVERAL
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
102 CHAPTER 3. MULTIPLE INTEGRALS T
Page 112 and 113:
104 CHAPTER 3. MULTIPLE INTEGRALS a
Page 114 and 115:
106 CHAPTER 3. MULTIPLE INTEGRALS l
Page 116 and 117:
108 CHAPTER 3. MULTIPLE INTEGRALS
Page 118 and 119:
110 CHAPTER 3. MULTIPLE INTEGRALS 3
Page 120 and 121:
112 CHAPTER 3. MULTIPLE INTEGRALS N
Page 122 and 123:
114 CHAPTER 3. MULTIPLE INTEGRALS w
Page 124 and 125:
116 CHAPTER 3. MULTIPLE INTEGRALS "
Page 126 and 127:
Page 128 and 129:
120 CHAPTER 3. MULTIPLE INTEGRALS
Page 130 and 131:
122 CHAPTER 3. MULTIPLE INTEGRALS E
Page 132 and 133:
Page 134 and 135:
126 CHAPTER 3. MULTIPLE INTEGRALS T
Page 136 and 137:
Page 138 and 139:
130 CHAPTER 3. MULTIPLE INTEGRALS E
Page 140 and 141:
132 CHAPTER 3. MULTIPLE INTEGRALS u
Page 142 and 143:
Page 144 and 145: 136 CHAPTER 4. LINE AND SURFACE INT
Page 196 and 197: 188 Bibliography Pogorelov, A.V., A
Page 198 and 199: 190 Appendix A: Answers and Hints t
Page 200 and 201: Appendix B We will prove the right-
Page 202 and 203: 194 Appendix B: Proof of the Right-
Page 204 and 205: Appendix C 3D Graphing with Gnuplot
Page 206 and 207: 198 Appendix C: 3D Graphing with Gn
Page 208 and 209: 200 Appendix C: 3D Graphing with Gn
Page 210 and 211: 202 GNU Free Documentation License
Page 218 and 219: Index Symbols D....................
Page 220 and 221: 212 Index iterated integral .......
show all

Michael Corral: Vector Calculus

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?