Multivariate Calculus - Bruce E. Shapiro

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6 LECTURE 1. CARTESIAN COORDINATES Figure 1.3: The plane in example 1.7 is sketched by first finding the three coordinate intercepts and then drawing a triangle to connect them. Figure 1.4: Left: the line x + y = 4 in the xy-plane. Right: cross section of the plane x + y = 4 with the plane z = 4. Revised December 6, 2006. Math 250, Fall 2006
LECTURE 1. CARTESIAN COORDINATES 7 Example 1.8 Find the equation of sphere that is tangent to the plane x + y = 4 with center at the point P = (1, 1, 4). (See the text, page 599 #28.) In two dimensions (the xy-plane), the equation x+y = 4 describes a line of slope −1 with y-intercept of 4. This line also crosses the x-axis (has x-intercept) at x = 4 (see figure 1.4.) In three dimensions, this equation x + y = 4 represents a vertical plane whose projection onto the xy-plane is the line x + y = 4. To find the equation of the desired sphere, consider the horizontal plane passing through P, namely, the plane z = 4. This plane sits parallel to and directly above the xy-plane, and from an illustrative point of view (see figure 1.4) looks just like the xy-plane. The center of the sphere will lay at (1, 1, 4), or at the point (1,1) of the equivalent xy-plane look-alike illustration. The cross section of the sphere with this plane is a circle of radius r (whose value is not yet determined) that is tangent to the line x + y = 4 in this plane. The point of tangency is point on the line that is closest to the point (1,1). By symmetry, this must be the midpoint of the x-intercept and y-intercept, which is ( 0 + 4 2 , 4 + 0 2 , 4 + 4 ) = (2, 2, 4) 2 The radius is the distance between the center at (1,1,4) and the point of tangency at (2,2,4): r = √ (1 − 2) 2 + (1 − 2) 2 + (4 − 4) 2 = √ 2 Hence the equation of the sphere is (x − 1) 2 + (y − 1) 2 + (z − 4) 2 = 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: LECTURE 1. CARTESIAN COORDINATES 5
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 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 67 and 68: LECTURE 8. FUNCTIONS OF TWO VARIABL
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LECTURE 8. FUNCTIONS OF TWO VARIABL
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Lecture 9 The Partial Derivative De
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LECTURE 9. THE PARTIAL DERIVATIVE 6
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Lecture 10 Limits and Continuity In
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LECTURE 10. LIMITS AND CONTINUITY 6
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LECTURE 10. LIMITS AND CONTINUITY 7
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Lecture 11 Gradients and the Direct
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LECTURE 11. GRADIENTS AND THE DIREC
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Lecture 12 The Chain Rule Recall th
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LECTURE 12. THE CHAIN RULE 83 The p
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LECTURE 12. THE CHAIN RULE 85 Examp
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LECTURE 12. THE CHAIN RULE 87 becau
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LECTURE 12. THE CHAIN RULE 89 Solut
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LECTURE 12. THE CHAIN RULE 91 Examp
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Lecture 13 Tangent Planes Since the
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LECTURE 13. TANGENT PLANES 95 Solut
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LECTURE 13. TANGENT PLANES 97 Multi
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LECTURE 13. TANGENT PLANES 99 Accor
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Lecture 14 Unconstrained Optimizati
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LECTURE 14. UNCONSTRAINED OPTIMIZAT
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Lecture 15 Constrained Optimization
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LECTURE 15. CONSTRAINED OPTIMIZATIO
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Lecture 16 Double Integrals over Re
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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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