Basic Multivariable Calculus 1993

MULTIVARIABLE
Preface
Contents
Algebra and Geometry of Euclidean Space
Vectors
The Standard Basis Vectors
The Vector Joining Two Points
Parametric Equation of a Line: Point-Direction Form
Parametric Equation of a Line: Point-Point Form
Inner Product, Length, and Distance
||j - i|| = 7(o -1)2+ (i-o)2 +(o-o)2 = 72. ♦
Angles and the Inner Product
Perpendicular Vectors
Cauchy-Schwarz Inequality
P l|a|Pa-
Orthogonal Projection
Triangle Inequality
Displacement and Velocity
Real-World Problems vs. Made-Up Problems
Exercises for
Definition of the Cross Product
The Cross Product
Geometry of 2 x 2 Determinants
Geometry of 3 x 3 Determinants
Equation of a Plane in Space
Distance from a Point to a Plane
Cauchy-Schwarz Inequality
H"iiH " WM
Triangle Inequality
Paths and Curves
Velocity Vector
Tangent Vector
Tangent Line to a Path
Exercises for
Differentiation
The Graph of a Function
Level Curves
Level Surfaces
Plotting Surfaces
Partial Differentiation
Limits—Intuitive Approach
Double and Single Limits in Two Variables
The e,6 Definition of Limit
Definition of Continuity
Composition
Continuity and Composition
Tangent Plane to a Graph
Differentiability
/(xo.yo) +
[ar, >
The Derivative
X^XO llx-Xoll
»/(*.)=
Differentiability and Continuity
Condition for Differentiability
The Chain Rule
Curves and Tangents
Curves on Graphs
Tangents to Curves on Surfaces
c'(t) = i/COi + A'Wj + kr(t)k.
The Chain Rule for Two Intermediate and Two Independent Variables
The Chain Rule for Three Intermediate and Two Independent Variables
The Chain Rule—General Case
Additional Derivative Rules
The Chain Rule—General Case
%
Additional Derivative Rules
The Gradient
The Chain Rule and Gradients
Gradients and Directional Derivatives
Tangent Plane to a Surface
Gradients and Tangent Planes
.•*>x’« *.*3(V.
c’A y J J ft «
Wm A< WMV AWWw cw '.MA V* tr CM/A MAM AMAA tA^A^KW.41 1 ■& !#. $ Ad’LA.AjAs -ar?W5 £Sh4Lft£
Uh
Equation of the Tangent Plane to a Level Surface
Implicit Differentiation and Partial Derivatives
50. /(z,a) = 17zM= 4=(i + j);(l>l)
V
Higher Derivatives and Extrema
Equality of Mixed Partial Derivatives
Equality of Mixed Partial Derivatives
0,
A Second Order Taylor Formula
z(y -yo) s ? + &2
g(s) = f(sx + (1 - s)x0, sy 4- (1 - s)yo)
/(0,0) = l, ^f(0,0) = l, ^(0,0) = 0,
^(0.0) = -l,
^(0,0)= cos(0 + 2-0) = 1,
§Z(0,0) =2cos(0 + 2-0) =2,
&0,0) = 0, &o,o) = 0,
Definition of Maxima and Minima
Critical Points
First Derivative Test
2y2 + y2 = 3y2 = 0,
-2y2 + y2 = -y2 = 0,
Absolute Maxima and Minima on Closed Intervals
f'(x) = 2x - 4 = 0.
Taylor's Formula Near a Critical Point
^(0,0) = ^(0,0) = 0.
The Shape of Graphs of Quadratic Functions
The Shape of Graphs of General Quadratic Functions
Second Derivative Test
9/ ^=xcosy-
x [2 - (x2 - y2)] = 0,
y [-2 - (x2 - y2)] - 0.
Exercises for
Critical Pointiest for Constrained Extrema
Method of Lagrange Multipliers
7(o,i) = /(o,-i) = -i, /(i,o) = 7(-i,0) = i,
Finding the Absolute Maximum and Minimum of f(x,y') on a Region D
Lagrange Multiplier Method in Space
4
Vector-Valued Functions
Differentiation Rules
0 = ~ [c(t). c(t)] = c'(t). C(i) + C(t). C'(i) = 2c(t) • C'(i);
Acceleration and Newton's Second Law
Kepler's Law
5- 4[ci(t) + c2(t)]
4[ci(f) xc2(t)]
Arc Length
Arc Length Differential
Arc Length in Rn
Vector Fields
F(x, y, z) = i — -g, r3 y, r3~z)- ♦
Flow Lines
§4.3
Divergence
^-(s2y) +
ci A
x 4-—(-2/) = 1 4-(-1) = 0. ♦ dx dy
Curl of a Vector Field
Curl of a Gradient
Divergence of a Curl
The Laplace Operator
Some Basic Identities of Vector Analysis
div(/F) = A(/F1) + |;(/r2) + ^(/F3).
4- CW = j^72i+t’ + k^ = 2
c(f) = [cos^(t)]i + [sing-(t)]j
Interlude: Where We Are Headed
Multiple Integrals
The Slice Method—Cava I ieri's Principle
Double and Iterated Integrals
b 1
Exercises for
Definition of the Double Integral
ILf' /fRf{x'y}dA’
Geometry of the Double Integral
Existence of Integrals
Properties of the Double Integral
Reduction to Iterated Integrals
IL
The Double Integral over a Region d
Continuity Implies Integrability
Iterated Integrals for Elementary Regions If £> is a region of type 1?
Mean Value Theorem for Single Integrals
Mean Value Theorem for Double Integrals
f fl (x + y2)dxdy
L fi + y2}dydx
l-^ILnx’v)dA-‘-
The Triple Integral over a Box
Reduction to Iterated Integrals
Triple Integrals by Iterated Integration
yyy # dxdydz
y yyy exyydxdydz--B = [°’ x 1°’ x I0’
//LfiV -
Double Integrals in Polar Coordinates
J J M.
The Gaussian Integral
Cylindrical Coordinates
Triple Integrals in Cylindrical Coordinates
Spherical Coordinates
Triple Integrals in Spherical Coordinates
HL
T
The Jacobian Determinant
Change of Variables Formula
ffD^x,^dxdy = //D ^x^u,v^’y^u,v^
Jacobians
Change of Variables in Triple Integrals
W-=
Average Value
Coordinates of the Center of Mass
Volume, Mass, Center of Mass, and Average Value for Regions in Space
17!,. = |
Moments of Inertia
/(x,y,z) =
1
= Gm//L
Integrals Over Curves and Surfaces
Line Integrals
/ F-ds= / tdt = 2tt2. ♦
. dz dy* dz* c<‘>=df1 + Si + 3k
Line Integrals—Differential Form Notation
Line Integrals of Gradient Fields
Independence of Parametrization
Line Integrals Along Geometric Curves
T(0 =
l|e'(0ll
Work
\r2 n/
Integrals of Scalar Functions Along Paths
Parametrized Surface
Tangent Plane
n = Ou x - (-1,0,1) = -i + k.
Area Element on a Graph
a/
Surface Area
= lfD l/WI \/l + LTM2 dudv = I" \f(u)171 + [/'(u)F dvdu
= 2?r f |/(u)l71 + [/'(^)]2 du, J a
Integral of a Scalar Function over a Surface
ZZ'^gZZ,^
Surface Integral
yy r • ds ~ yy * - x
Orientation
The Surface Integral for Graphs
1 1 _ 1 _ 1 _ 4
2 + 2 + 3 3 ”3'
Independence of Parametrization
Area of a Shadow
Surface Integrals for Fluid Flow
The Integral Theorems of Vector Analysis
Orienting the Boundary Curve
Green's Theorem
Area of a Region
Vector Form of Green's Theorem
i k
a d_
Q 0
_ 1 1 1 1 - 1
2 4 3 + 6 ~ 12'
Gauss' Divergence Theorem in the Plane
IL (& - dxdvIL(%+%)dxdy=IL di',tdxdy’
f[ Wl. . .
/ “x+ <9^- }dy
Stokes' Theorem
Circulation and Curl
-* fl,-■—fl,-® ds-/Z LlJ—%//**■
Gauss' (Divergence) Theorem
= yyy*
If
fl III
Divergence and Flux
Gauss' Law
v2 = -p.
Path Independence
Conservative Vector Fields
Curl and Gradient
Cross-Derivative Test in the Plane
Antiderivatives of Vector Fields
Surface Independence
yy g » ds—yyy =o,
Divergence and Curl
Epilogue:
Where Do We Go from Here?
Practice Examination 1
Practice Examination 2
Answers to Odd-Numbered Exercises
1.2 The Inner Product and Distance
1.4 The Cross Product and Planes
1.5 n-dimensional Euclidean Space
1<6 Curves in the Plane and in Space
Review Exercises (Chapter 1)
Chapter 2 Differentiation
2.1 Graphs and Level Surfaces
2.2 Partial Derivatives and Continuity
2.5 Gradients and Directional Derivatives
Chapter 3
Higher Derivatives and Extrema
3.1 Higher Order Partial Derivatives
13. 4(l|v||2) = —(v-v)=2v-= 2va = 0
15. 6129 seconds
1.
5.
9.
11.
13.
2v/57r
3.
7.
2(2v^ - 1)
5. F = (2y,x)
9. The flow lines are concentric circles
11. The flow lines for t > 0:
15. c'(t) = ( cost, — sin t,e f) = F(c(t))
5. div V > 0 in the first and third quadrants,
Review Exercises (Chapter 4)
5.2 The Double Integral Over a Rectangle
sin(l)
9
7- -e9)" 5(e4 ’ e}
16/9
76/3 25/6
2 e
5.3 The Double Integral Over Regions
5.5 Change of Variables
5.6 Applications of Multiple Integrals
Review Exercises (Chapter 5)
Chapter 6
Integrals Over Curves and Surfaces
6.1 Line Integrals
6.2 Parametrized Surfaces
6.3 Area of a Surface
6.4 Surface Integrals
Review Exercises (Chapter 6)
7.3 Gauss' Theorem
=yyy dxdydz 4- yyy
7.4 Path Independence and the Fundamental Theorems of Calculus
Practice Exam 1
Practice Exam 2
Index