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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 www.abbymath.com Abby Brown ~ 11/2003
