Calculus Day 4 - November 1, 2007
Topics: Vector calculus. Contour, surface, volume integrals. Divergence theorem, Green’s Theorem, Stokes’ Theorem. Change of variables, eg polar and
spherical coordinates.
Problems:
J07#3, J06#2, S02#5, J02#5, J05#4, J92#5, J96#4, J97#2.
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Divergence Theorem Let F~ : R3 → R3 be a C 1 vector field, and let V be a
region bounded by a simple closed surface S with unit outward normal ~n. Then
Z
Z
F~ · ~ndS =
∇ · F~ dV.
S
V
Green’s Theorem Let F~ = (P (x, y), Q(x, y)) be a C 1 vector field, let C be
a simple closed curve in R2 , and let S be the region enclosed by C. Then
I
Z
P dx + Qdy = (Qx − Py )dxdy.
C
S
This can also be written C F~ · d~r =
H
R
S
(k̂ · ∇ × F~ )dxdy.
Stokes’ Theorem Let Γ be any simple closed curve in R3 , and S any surface
which has Γ as its boundary. Let F~ : R3 → R3 be C 1 on S. Then
Z
Z
~
F · d~x = (∇ × F~ ) · ~ndS,
Γ
S
where direction of the unit normal ~n is determined by the right-hand rule.
If S is on the xy-plane, this reduces to Green’s Theorem.
Vector Identities
This list is NOT exhaustive!
∇ × (∇φ) = 0
∇ · (∇ × F~ ) = 0
∇ · (∇φ) = ∆φ
∇ · (φF~ ) = φ∇ · F~ + F~ · ∇φ
Polar coordinates
x = r cos θ
,
y = r sin θ
θ ∈ [0, 2π],
∂(x, y)
=r
∂(r, θ)
Spherical coordinates
x = r sin φ cos θ
y = r sin φ sin θ ,
z = r cos φ
φ ∈ [0, π]
,
θ ∈ [0, 2π]
1
∂(x, y, z)
= r2 sin φ
∂(r, φ, θ)