Michael Corral: Vector Calculus

More documents

Recommendations

Info

50 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE ☛ ✟ ✡Exercises ✠ A For Exercises 1-4, find the (a) cylindrical and (b) spherical coordinates of the point whose Cartesian coordinates are given. 1. (2,2 √ 3,−1) 2. (−5,5,6) 3. ( √ 21,− √ 7,0) 4. (0, √ 2,2) For Exercises 5-7, write the given equation in (a) cylindrical and (b) spherical coordinates. 5. x 2 +y 2 +z 2 = 25 6. x 2 +y 2 = 2y 7. x 2 +y 2 +9z 2 = 36 B 8. Describe the intersection of the surfaces whose equations in spherical coordinates areθ= π 2 andφ= π 4 . 9. Showthatfora0,theequationρ=2asinφ cosθ insphericalcoordinatesdescribes a sphere centered at (a,0,0) with radius|a|. C 10. Let P=(a,θ,φ) be a point in spherical coordinates, with a>0and 0
1.8 <strong>Vector</strong>-Valued Functions 51 1.8 <strong>Vector</strong>-Valued Functions Now that we are familiar with vectors and their operations, we can begin discussing functions whose values are vectors. Definition 1.10. A vector-valued function of a real variable is a rule that associates a vector f(t) with a real number t, where t is in some subset D of 1 (called the domain of f). We write f : D→ 3 to denote that f is a mapping of D into 3 . For example, f(t)=ti+t 2 j+t 3 k is a vector-valued function in 3 , defined for all real numbers t. We would write f :→ 3 . At t=1the value of the function is the vector i+j+k, which in Cartesian coordinates has the terminal point (1,1,1). A vector-valued function of a real variable can be written in component form as or in the form f(t)= f 1 (t)i+ f 2 (t)j+ f 3 (t)k f(t)=(f 1 (t), f 2 (t), f 3 (t)) for some real-valued functions f 1 (t), f 2 (t), f 3 (t), called the component functions of f. The first form is often used when emphasizing that f(t) is a vector, and the second form is usefulwhenconsideringjusttheterminalpointsofthevectors. Byidentifyingvectors withtheirterminalpoints,acurveinspacecanbewrittenasavector-valuedfunction. Example 1.35. Define f :→ 3 by f(t)=(cost,sint,t). This is the equation of a helix (see Figure 1.8.1). As the value of t increases, the terminal points of f(t) trace out a curve spiraling upward. For each t, the x- and y-coordinates of f(t) are x=cost and y=sint, so x 2 +y 2 = cos 2 t+sin 2 t=1. Thus, the curve lies on the surface of the right circular cylinder x 2 +y 2 = 1. f(2π) f(0) x 0 z Figure 1.8.1 It may help to think of vector-valued functions of a real variable in 3 as a generalization of the parametric functions in 2 which you learned about in single-variable calculus. Much of the theory of real-valued functions of a single real variable can be applied to vector-valued functions of a real variable. Since each of the three component functions are real-valued, it will sometimes be the case that results from singlevariable calculus can simply be applied to each of the component functions to yield a similar result for the vector-valued function. However, there are times when such generalizations do not hold (see Exercise 13). The concept of a limit, though, can be extended naturally to vector-valued functions, as in the following definition. y
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 18 and 19: 10 CHAPTER 1. VECTORS IN EUCLIDEAN
Page 74 and 75: 66 CHAPTER 2. FUNCTIONS OF SEVERAL
Page 108 and 109:
100 CHAPTER 2. FUNCTIONS OF SEVERAL
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 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
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 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Index Symbols D....................
Page 220 and 221:
212 Index iterated integral .......
show all

Michael Corral: Vector Calculus

Create successful ePaper yourself

Delete template?

Save as template?