Calculus D Notes

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Chapter 16
Vector Calculus
16-1 & 16-5 Vector Fields, Curl, and Divergence
16-2 Line Integrals
16-3 The Fundamental Theorem for Line Integrals
16-4 Green’s Theorem
16-6 Parametric Surfaces
16-7 Surface Integrals
16-9 The Divergence Theorem
16-8 Stokes’s Theorem
16-10 Summary
The following notes are for the Calculus D (SDSU Math 252)
classes I teach at Torrey Pines High School. I wrote and
modified these notes over several semesters. The
explanations are my own; however, I borrowed several
examples and diagrams from the textbooks* my classes used
while I taught the course. Over time, I have changed some
examples and have forgotten which ones came from which
sources. Also, I have chosen to keep the notes in my own
handwriting rather than type to maintain their informality
and to avoid the tedious task of typing so many formulas,
equations, and diagrams. These notes are free for use by my
current and former students. If other calculus students and
teachers find these notes useful, I would be happy to know
that my work was helpful. - Abby Brown
SDUHSD Calculus III/D
SDSU Math 252
Abby Brown
www.abbymath.com
San Diego, CA
*Calculus: Early Transcendentals, 6th & 4th editions, James Stewart, ©2007 & 1999
Brooks/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.)
*Calculus, 5th edition, Roland E. Larson, Robert P. Hostetler, & Bruce H. Edwards, ©1994
D. C. Heath and Company, ISBN 0-669-35335-3.
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Integration Summary
Scalar Functions
z
zz
b
interval length =
A=
dx
a
A=
V =
dA
R
V = ∫∫∫ dV
z
zz
b
a
f dx
f dA
z
= ds
arc length = s
=
surface area = S
R
mass
zz
“curtain” area or mass =
C
dS
mass of surface lamina =
S
= ∫∫∫ f dV
z
zz
C
f ds
f dS
S
Note: Integral represents “mass” if f is a density function.
E
E
Vector Functions
work = ∫ F ⋅ dr
flux
C
zz
zz
zz
zz
=
F ⋅ dS
S
= ∫ F ⋅ T ds
=
C
= ∫ F ⋅ r′(t ) dt
=
a
= ∫ P dx + Q dy + R dz ← differential form
r ′(t )
r ′(t )
N=
∇G
∇G
F ⋅ N dS
S
b
T=
F ⋅∇G dA
R
C
=
F ⋅ (ru × rv ) dA ← parametric form
D
Elements of Integration
dA = dy dx, r dr d2, du dv
dV = dz dy dx, r dz dr d2, D2sinN dD dN d2
dr = T ds = r ′ ( t ) dt
dS = N dS = ∇G dA
ds = r ′( t ) dt = [ x ′ ( t )]2 + [ y ′ ( t )]2 + [ z ′ ( t )]2 dt
dS = ∇G dA = [ g x ( x , y )]2 + [ g y ( x , y )]2 + 1 dA where z = g ( x , y ) and G ( x , y , z ) = z − g ( x , y )
= ru × rv dA where S is given by r ( u, v ) ← parametric form
F
C
Closed
z
C
Conservative
(› a potential function f such that F=Lf)
F ⋅ dr = 0
Note: Green’s, Stokes’s, and
Fundamental Theorem also
apply in this case.
Not Conservative
Green’s Theorem (2D)
∫
Fundamental Theorem of Line Integrals
Not
Closed
z
F ⋅ dr = f ( x ( b), y ( b), z ( b))
− f ( x ( a ), y ( a ), z ( a ))
C
where F = ∇ f
If S is closed: Divergence Theorem
∫∫ F ⋅ dS = ∫∫∫ div F dV
S
E
C
z
C
F ⋅ dr = ∫∫
R
∂Q ∂P
−
dA
∂x ∂y
z
Stokes’s Theorem (3D)
∫
C
F ⋅ dr = ∫∫ curl F ⋅ dS
S
b
F ⋅ dr = F ⋅ r ′( t ) dt
a
Complete the line integral
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Abby Brown ~ 11/2003