Post on 07-Feb-2018
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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