Line Integrals
} In 2 dimensions: Let C be a smooth curve given by
r : R Ñ R2 (from 1D to 2D)
.
rptq “ p xptq , yptq q
} In 3 dimensions: Let C be a smooth curve given by
r : R Ñ R3 (from 1D to 3D)
rptq “ p xptq , yptq , zptq q
Line integral
Scalar function
f returns a number
Notation
ª
f ds
Formula
ª
C
Vector field
F returns a vector
ª
C
F ¨ dr
(work done by F along C)
} Special case: F “ rf
(conservative field)
C
ª
C
ª
f prptqq|r1 ptq| dt
F ¨ T ds “
C
ª
C
F prptqq ¨ r1 ptq dt
`
˘
`
˘
rf ¨ dr “ f rpbq ´ f rpaq
= f (end) - f (start)
Surface Integrals
} Let S be a surface with vector equation r : R2 Ñ R3
(from 2D to 3D)
rpu, vq “ p xpu, vq , ypu, vq , zpu, vq q
.
Surface integral
Scalar function
f : R3 Ñ R
f returns a number
Vector field
F : R3 Ñ R3
F “ pP, Q, Rq
returns a vector
Notation
º
º
f px, y, zqdS
S
f prpu, vqq|ru ˆ rv | dA
D
º
F ¨ dS
S
(flux of F across S)
If S given by z “ gpx, yq
Formula
º
D
F ¨ n dS “
º
D
F ¨ pru ˆ rv q dA
º
c
f px, y, gpx, yqq 1 `
D
´ ¯2
Bz
Bx
`
´ ¯2
Bz
By
˙
º ˆ
Bg
Bg
´P
´ Q ` R dA
Bx
By
D
dA
Important Theorems
.
Green’s Theorem
˙
º ˆ
BQ BP
F ¨ dr “
P dx ` Qdy “
´
dA
Bx
By
C
C
ª
ª
D
Recall: F “ pP, Qq is a vector field
ª
C
Stokes’ Theorem
º
F ¨ dr “
curlF ¨ dS
S
C = boundary of S (+ orientation)
Remember
curlF “ ˆ F
divF “ ¨ F
º
S
Divergence Theorem
¡
F ¨ dS “
divF dV
E
.
.
Regular Integrals
.
Double Integrals
Green's
Theorem
Parametrize
the curve C,
i.e. find r(t)
"
rectangular
polar
Parametrize
the surface S,
i.e. find r(u,v)
Line Integrals
} Scalar function f :
} Vector field F :
Special cases:
˚ F is conservative
˚ C is a closed path
Fund. Thm.
of line
integrals
≥
$
& rectangular
Triple Integrals
cylindrical
%
spherical
Surface Integrals
≥
C
f dr
C F ¨ dr
} Scalar function f :
} Vector field F :
Stokes'
Theorem
Special cases:
¥
˚ curlF ¨ dS
¥
S
¥
f dS
S
F ¨ dS
S
˚ S is a “closed” surface
Divergence
Theorem