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Chapter 12Vectors and theGeometry of Space
12-1 & 12-2 Vectors and 3-Dimensional Coordinates12-3 The Dot Product12-4 The Cross Product12-5 Equations of Lines and Planes12-6 Cylinders and Quadric Surfaces12-7 Cylindrical and Spherical Coordinates (4th ed.)
SDUHSD
Abby Brown
Calculus III/DSDSU Math 252
www.abbymath.comSan Diego, CA
The following notes are for the Calculus D (SDSU Math 252)classes I teach at Torrey Pines High School. I wrote andmodified these notes over several semesters. Theexplanations are my own; however, I borrowed severalexamples and diagrams from the textbooks* my classes usedwhile I taught the course. Over time, I have changed someexamples and have forgotten which ones came from whichsources. Also, I have chosen to keep the notes in my ownhandwriting rather than type to maintain their informalityand to avoid the tedious task of typing so many formulas,equations, and diagrams. These notes are free for use by mycurrent and former students. If other calculus students andteachers find these notes useful, I would be happy to knowthat my work was helpful. - Abby Brown
* , 6th & 4th editions, James Stewart, ©2007 & 1999Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2.(Chapter, section, page, and formula numbers refer to the 6th edition of this text.)
, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards,
Calculus: Early Transcendentals
*Calculus ©1994D. C. Heath and Company, ISBN 0-669-35335-3.
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Note: To find the distance between two parallel planes, choose any point on one of the planes and call that Q. Use Q and the other plane as when finding the distance between a point and a plane.
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Chapter 13Vector-Valued Functions
13-1 Vector Functions and Space Curves13-2 Derivatives and Integrals of Vector Functions
Unit Tangent Vectors13-4 Velocity and Acceleration13-3 Tangent and Normal Vectors
Arc Length and Curvature
The following notes are for the Calculus D (SDSU Math 252)classes I teach at Torrey Pines High School. I wrote andmodified these notes over several semesters. Theexplanations are my own; however, I borrowed severalexamples and diagrams from the textbooks* my classes usedwhile I taught the course. Over time, I have changed someexamples and have forgotten which ones came from whichsources. Also, I have chosen to keep the notes in my ownhandwriting rather than type to maintain their informalityand to avoid the tedious task of typing so many formulas,equations, and diagrams. These notes are free for use by mycurrent and former students. If other calculus students andteachers find these notes useful, I would be happy to knowthat my work was helpful. - Abby Brown
SDUHSD
Abby Brown
Calculus III/DSDSU Math 252
www.abbymath.comSan Diego, CA
* , 6th & 4th editions, James Stewart, ©2007 & 1999Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2.(Chapter, section, page, and formula numbers refer to the 6th edition of this text.)
, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards,
Calculus: Early Transcendentals
*Calculus ©1994D. C. Heath and Company, ISBN 0-669-35335-3.
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Chapter 14Partial Derivatives
14-1 Functions of Several Variables14-2 Limits and Continuity14-3 Partial Derivatives14-5 The Chain Rule14-6 Directional Derivatives
GradientsTangent Planes and Normal Lines
14-7 Maximum and Minimum Values
The following notes are for the Calculus D (SDSU Math 252)classes I teach at Torrey Pines High School. I wrote andmodified these notes over several semesters. Theexplanations are my own; however, I borrowed severalexamples and diagrams from the textbooks* my classes usedwhile I taught the course. Over time, I have changed someexamples and have forgotten which ones came from whichsources. Also, I have chosen to keep the notes in my ownhandwriting rather than type to maintain their informalityand to avoid the tedious task of typing so many formulas,equations, and diagrams. These notes are free for use by mycurrent and former students. If other calculus students andteachers find these notes useful, I would be happy to knowthat my work was helpful. - Abby Brown
SDUHSD
Abby Brown
Calculus III/DSDSU Math 252
www.abbymath.comSan Diego, CA
* , 6th & 4th editions, James Stewart, ©2007 & 1999Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2.(Chapter, section, page, and formula numbers refer to the 6th edition of this text.)
, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards,
Calculus: Early Transcendentals
*Calculus ©1994D. C. Heath and Company, ISBN 0-669-35335-3.
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Example: Related Rates The radius of a right circular cylinder is increasing at the rate of 6 inches per minute and the height is decreasing at the rate of 4 inches per minute. What is the rate of change of the volume and surface area when the radius is 12 inches and the height is 36 inches?
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Chapter 15Multiple Integrals
15-2 & 15-3 Iterated Integrals15-1, 15-2, & 15-3 Double Integrals and Volume15-4 Double Integrals in Polar Coordinates15-6 Surface Area (4th ed.)15-6 Triple Integrals15-7 Triple Integrals in Cylindrical Coordinates15-8 Triple Integrals in Spherical Coordinates
The following notes are for the Calculus D (SDSU Math 252)classes I teach at Torrey Pines High School. I wrote andmodified these notes over several semesters. Theexplanations are my own; however, I borrowed severalexamples and diagrams from the textbooks* my classes usedwhile I taught the course. Over time, I have changed someexamples and have forgotten which ones came from whichsources. Also, I have chosen to keep the notes in my ownhandwriting rather than type to maintain their informalityand to avoid the tedious task of typing so many formulas,equations, and diagrams. These notes are free for use by mycurrent and former students. If other calculus students andteachers find these notes useful, I would be happy to knowthat my work was helpful. - Abby Brown
SDUHSD
Abby Brown
Calculus III/DSDSU Math 252
www.abbymath.comSan Diego, CA
* , 6th & 4th editions, James Stewart, ©2007 & 1999Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2.(Chapter, section, page, and formula numbers refer to the 6th edition of this text.)
, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards,
Calculus: Early Transcendentals
*Calculus ©1994D. C. Heath and Company, ISBN 0-669-35335-3.
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How do we decide dxdy or dydx?Consider both (1) the shape of the region and (2) the integrand.Usually one order of integration is easier than the other. To switch the order of integration, sketch the region determined by the limits and use the graph to help you write new limits.
1 2 3
1
x y4
x 2 y
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Given a surface f(x,y), we can approximate the volume under the surface over a given region R in the xy-plane. Break up the region into squares, calculate the height at a corresponding point in each square, calculate the volume of each rectangular prism, and add all of the volumes together. Then change the approximation to an infinite number of prisms by taking the limit.
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Chapter 16Vector Calculus
16-1 & 16-5 Vector Fields, Curl, and Divergence16-2 Line Integrals16-3 The Fundamental Theorem for Line Integrals16-4 Green’s Theorem16-6 Parametric Surfaces16-7 Surface Integrals16-9 The Divergence Theorem16-8 Stokes’s Theorem16-10 Summary
The following notes are for the Calculus D (SDSU Math 252)classes I teach at Torrey Pines High School. I wrote andmodified these notes over several semesters. Theexplanations are my own; however, I borrowed severalexamples and diagrams from the textbooks* my classes usedwhile I taught the course. Over time, I have changed someexamples and have forgotten which ones came from whichsources. Also, I have chosen to keep the notes in my ownhandwriting rather than type to maintain their informalityand to avoid the tedious task of typing so many formulas,equations, and diagrams. These notes are free for use by mycurrent and former students. If other calculus students andteachers find these notes useful, I would be happy to knowthat my work was helpful. - Abby Brown
SDUHSD
Abby Brown
Calculus III/DSDSU Math 252
www.abbymath.comSan Diego, CA
* , 6th & 4th editions, James Stewart, ©2007 & 1999Brooks/Cole Publishing Company, ISBN 0-495-01166-5 & 0-534-36298-2.(Chapter, section, page, and formula numbers refer to the 6th edition of this text.)
, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards,
Calculus: Early Transcendentals
*Calculus ©1994D. C. Heath and Company, ISBN 0-669-35335-3.
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www.abbymath.comAbby Brown ~ 11/2003
d ds t dtd dS G dA
r T rS N= = ′= = ∇
( )
T rr
N
=′′
=∇∇
( )( )tt
GG
Integration SummaryScalar Functions
interval length = arc length = “curtain” area or mass = dxa
bz A f dxa
b= z s ds
C= z f ds
Cz surface area = mass of surface lamina =A dA
R
= zz V f dAR
= zz S dSS
= zz f dSSzz
mass Note: Integral represents “mass” if f is a density function.∫∫∫=E
dVV ∫∫∫=E
dVf
Vector Functionswork flux
∫∫
∫∫
←++=
′⋅=
⋅=
⋅=
C
b
a
C
C
dzRdyQdxP
dtt
ds
d
)(rF
TF
rF = ⋅
= ⋅
= ⋅∇
= ⋅ × ←
zzzzzzzz
F S
F N
F
F r r
d
dS
G dA
dA
S
S
R
u vD
( ) parametric form
Elements of Integration
dA = dy dx, r dr d2, du dvdV = dz dy dx, r dz dr d2, D2sinN dD dN d2
ds t dt x t y t z t dt
dS G dA g x y g x y dA z g x y G x y z z g x y
dA u vx y
u v
= ′ = ′ + ′ + ′
= ∇ = + + = = −
= × ←
r
r r r
( ) [ ( )] [ ( )] [ ( )]
[ ( , )] [ ( , )] ( , ) ( , , ) ( , )
( , )
2 2 2
2 2 1
where and
where is given by parametric formS
C F Conservative(› a potential function f such that F=Lf) Not Conservative
Closed F r⋅ =z dC
0 Note: Green’s, Stokes’s, andFundamental Theorem alsoapply in this case.
Green’s Theorem (2D) Stokes’s Theorem (3D)
∫∫∫ ∂∂
−∂∂
=⋅R
CdA
yP
xQdrF ∫ ∫∫ ⋅=⋅
CS
dd SFcurlrF
NotClosed
Fundamental Theorem of Line Integrals
F r⋅ =z d f x b y b z bC
( ( ), ( ), ( ))
F r F r⋅ = ⋅ ′z zd t dt
Ca
b( )
Complete the line integral
If S is closed: Divergence Theorem ∫∫∫∫∫ =⋅ES
dVd FSF div
differential form
−= ∇
f x a y a z af
( ( ), ( ), ( ))where F
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