Michael Corral: Vector Calculus

More documents

Recommendations

Info

66 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES Example 2.4. The domain of the function f(x,y,z)=e x+y−z is all of 3 , and the range of f is all positive real numbers. A function f(x,y) defined in 2 is often written as z= f(x,y), as was mentioned in Section 1.1, so that the graph of f(x,y) is the set{(x,y,z) : z= f(x,y)} in 3 . So we see thatthisgraph isasurfacein 3 , sinceitsatisfiesanequationoftheform F(x,y,z)=0 (namely, F(x,y,z)= f(x,y)−z). The traces of this surface in the planes z=c, where c varies over, are called the level curves of the function. Equivalently, the level curves are the solution sets of the equations f(x,y)=c, for c in. Level curves are often projected onto the xy-planeto give an idea of the various “elevation”levels of the surface (as is done in topography). Example 2.5. The graph of the function f(x,y)= sin√ x 2 +y 2 √ x 2 +y 2 is shown below. Note that the level curves (shown both on the surface and projected onto the xy-plane) are groups of concentric circles. 1 0.8 0.6 0.4 z 0.2 0 -0.2 -0.4 -10 -5 y 0 5 10 10 5 0 -5 x -10 Figure 2.1.1 The function f(x,y)= sin√ x 2 +y 2 √ x 2 +y 2 You may be wondering what happens to the function in Example 2.5 at the point (x,y) = (0,0), since both the numerator and denominator are 0 at that point. The function is not defined at (0,0), but the limit of the function exists (and equals 1) as (x,y) approaches (0,0). We will now state explicitly what is meant by the limit of a function of two variables.
2.1 Functions of Two or Three Variables 67 Definition 2.1. Let (a,b) be a point in 2 , and let f(x,y) be a real-valued function defined on some set containing (a,b) (but not necessarily defined at (a,b) itself). Then we say that the limit of f(x,y) equals L as (x,y) approaches (a,b), written as if given anyǫ> 0, there exists aδ>0such that |f(x,y)−L|
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: 16 CHAPTER 1. VECTORS IN EUCLIDEAN
Page 76 and 77: 68 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:
118 CHAPTER 3. MULTIPLE INTEGRALS a
Page 128 and 129:
120 CHAPTER 3. MULTIPLE INTEGRALS
Page 130 and 131:
122 CHAPTER 3. MULTIPLE INTEGRALS E
Page 132 and 133:
124 CHAPTER 3. MULTIPLE INTEGRALS 3
Page 134 and 135:
126 CHAPTER 3. MULTIPLE INTEGRALS T
Page 136 and 137:
128 CHAPTER 3. MULTIPLE INTEGRALS 3
Page 138 and 139:
130 CHAPTER 3. MULTIPLE INTEGRALS E
Page 140 and 141:
132 CHAPTER 3. MULTIPLE INTEGRALS u
Page 142 and 143:
134 CHAPTER 3. MULTIPLE INTEGRALS a
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

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

Delete template?

Save as template?