Multivariate Calculus - Bruce E. Shapiro
Multivariate Calculus - Bruce E. Shapiro Multivariate Calculus - Bruce E. Shapiro
4 LECTURE 1. CARTESIAN COORDINATES Example 1.5 Suppose that two spheres of equal radii have their centers at P = (−3, 1, 2) Q = (5, −3, 6) Find the equations of the two spheres if the two spheres are just touching (tangent) at precisely one point. (See the text, page 599 #26.) Since the two spheres are tangent at a single point, the radius is one half the distance between the two centers (think of two bowling balls that are just touching one another). Hence 2r = |P Q| = √ (−3 − 5) 2 + (1 − −3) 2 + (2 − 6) 2 = √ (−8) 2 + (4) 2 + (−4) 2 = √ 64 + 16 + 16 = √ 96 and therefore r = 1 √ 96 ⇒ r 2 = 96 2 4 = 24 The equation of the sphere centered at P = (−3, 1, 2)is (x + 3) 2 + (x − 1) 2 + (x − 2) 2 = 24 while the equation of the sphere whose center is Q = (5, −3, 6) is (x − 5) 2 + (y + 3) 2 + (z − 6) 2 = 24. Theorem 1.1 Midpoint formula. The coordinates of a point halfway between the points P = (x 1 , y 1 , z 1 ) and Q = (x 2 , y 2 , z 2 ) are given by the formula ( x1 + x 2 (m 1 , m 2 , m 3 ) = , y 1 + y 2 , z ) 1 + z 2 2 2 2 (1.7) Example 1.6 Find the equations of the sphere that has the line segment between (-2,3,6) and (4,-1,5) as its diameter. The center of the sphere must be the center of the line segment between the two points (-2, 3, 6) and (4, -1, 5), ( −2 + 4 C = , 3 + −1 , 6 + 5 ) = (1, 1, 5.5) 2 2 2 The diameter of the sphere is the length of the line segment, d = √ (−2 − 4) 2 + (3 − −1) 2 + (6 − 5) 2 = √ 36 + 16 + 1 = √ 53 The radius is one-half the diameter, r = √ 53/2 The equation of the sphere is then (x − 1) 2 + (y − 1) 2 + (z − 5.5) 2 = 53/4. Revised December 6, 2006. Math 250, Fall 2006
LECTURE 1. CARTESIAN COORDINATES 5 Definition 1.4 A linear equation is any equation of the form Ax + By + Cz = D (1.8) where A, B, C and D are constants and A, B and C are not all zero (D may be zero). Theorem 1.2 An equation is linear if and only if it is the equation of a plane, i.e., linear equations are equations of planes, and all equations of planes are linear equations. Theorem 1.3 Properties of a linear equation. Suppose that Ax+By+Cz = D. Then 1. If A ≠ 0, the x-intercept is at D/A. If A = 0, the plane is parallel to the x-axis. 2. If B ≠ 0, The y-intercept is at D/B. If B = 0, the plane is parallel to the y-axis. 3. If C ≠ 0, The z-intercept is at D/C. If C = 0, the plane is parallel to the z-axis If D ≠ 0then the equation of a plane is which we can rewrite as Let Then A D x + B D y + C D z = 1 x D/A + y D/B + a = D/A b = D/B c = D/C z D/C = 1 x a + y b + z c = 1 where a is the x-intercept, b is the y-intercept, and c is the z-intercept. Example 1.7 Sketch the plane −3x + 2y + (3/2)z = 6 The x-intercept is at 6/(-3)=-2, which is the point (-2, 0, 0) The y-intercept is at 6/2=3, which is the point (0, 3, 0) The z-intercept is at 6/(3/2)=4, which is the point (0, 0, 4) The plane is sketched in figure 1.3. 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: LECTURE 1. CARTESIAN COORDINATES 3
- Page 19 and 20: LECTURE 1. CARTESIAN COORDINATES 7
- 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: LECTURE 4. LINES AND CURVES IN 3D 3
- Page 47 and 48: 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 53 and 54: LECTURE 5. VELOCITY, ACCELERATION,
- Page 55 and 56: 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
LECTURE 1. CARTESIAN COORDINATES 5<br />
Definition 1.4 A linear equation is any equation of the form<br />
Ax + By + Cz = D (1.8)<br />
where A, B, C and D are constants and A, B and C are not all zero (D may be<br />
zero).<br />
Theorem 1.2 An equation is linear if and only if it is the equation of a plane,<br />
i.e., linear equations are equations of planes, and all equations of planes are linear<br />
equations.<br />
Theorem 1.3 Properties of a linear equation. Suppose that Ax+By+Cz = D.<br />
Then<br />
1. If A ≠ 0, the x-intercept is at D/A. If A = 0, the plane is parallel to the<br />
x-axis.<br />
2. If B ≠ 0, The y-intercept is at D/B. If B = 0, the plane is parallel to the<br />
y-axis.<br />
3. If C ≠ 0, The z-intercept is at D/C. If C = 0, the plane is parallel to the<br />
z-axis<br />
If D ≠ 0then the equation of a plane is<br />
which we can rewrite as<br />
Let<br />
Then<br />
A<br />
D x + B D y + C D z = 1<br />
x<br />
D/A +<br />
y<br />
D/B +<br />
a = D/A<br />
b = D/B<br />
c = D/C<br />
z<br />
D/C = 1<br />
x<br />
a + y b + z c = 1<br />
where a is the x-intercept, b is the y-intercept, and c is the z-intercept.<br />
Example 1.7 Sketch the plane −3x + 2y + (3/2)z = 6<br />
The x-intercept is at 6/(-3)=-2, which is the point (-2, 0, 0)<br />
The y-intercept is at 6/2=3, which is the point (0, 3, 0)<br />
The z-intercept is at 6/(3/2)=4, which is the point (0, 0, 4)<br />
The plane is sketched in figure 1.3. <br />
Math 250, Fall 2006 Revised December 6, 2006.
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