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MATH 332: Vector Analysis
Formulas
Vector Algebra
x 1 = x, x 2 = y, x 3 = z
e 1 = i, e 2 = j, e 3 = k
ei · ej = δij
ei × ej = εijkek
A = Aiei
A · B = AiBi
A · B = B · A
|A|
2 = A · A = AiAi
A × B = εijkAj Bkei =
i j k
A 1 A 2 A 3
B 1 B 2 B 3
(A × B)i = εijkAj Bk
A × A = 0
[A, B, C] = A · (B × C) = εijkAiBj Ck
[A, B, C] = [B, C, A] = [C, A, B]
Line parallel to A:
xi = x(0) i + Ait
t =
xi − x(0) i
Ai
, Ai 6 = 0
Plane orthogonal to N
Ni(xi − x(0) i) = 0
Identities:
A × (B × C) = B(A · C) − C(A · B)
Tensors:
δij = δji
δii = 3
δij Aj = Ai
δij AiBj = AiBi = A · B
εijk = −εjik = −εikj = −εkji
εijk = εjki = εkij
εijj = εjij = εjji = 0
εijkδij = εijkδik = εijkδjk = 0
εijkAj Ak = εijkAiAk = εijkAiAj = 0
δ
i j =^ δ
j i =^ δ
ij = δij
ε
ijk = εijk
εijkε
mnl = δ
m i δ
n j δ
l k +^ δ
m j δ
n k δ
l i +^ δ
m k δ
n i δ
l j
− δ
m i δ
n k δ
l j −^ δ
m j δ
n i δ
l k −^ δ
m k δ
n j δ
l i
εijkε
mnk = δ
m i δ
n j −^ δ
m j δ
n i
εijkε
mjk = 2δ
m i
εijkε
ijk = 6
Vector Functions
Position
R = xiei = xi + yj + zk
Velocity
v =
dR
dt
Acceleration
a =
dv
dt
d
2 R
dt
2
Arc Length
ds = |v|dt
s(t) =
∫ (^) t
t 0
|v(τ )| dτ
Speed
|v| =
ds
dt
Tangent
T =
v
|v|
Curvature
k =
|v|
dT
dt
|v × a|
|v|^3
Radius of curvature
ρ =
k
Principal Normal
N =
k|v|
dT
dt
Binormal
B = T × N
Torsion
τ =
|v|
B ·
dN
dt
Scalar and Vector Fields
Partial derivatives
∂i =
∂xi
∂ 1 = ∂x, ∂ 2 = ∂y, ∂ 3 = ∂z
Nabla (Del) Operator
∇ = ei∂i = i∂x + j∂y + k∂z
Gradient
grad f = ∇f = ei∂if
∇if = ∂if
Directional derivative
df
ds
dR
ds
· gradf =
dxi
ds
∂if
Flow curves dxi
ds
= βFi
∫ Q
P
∇ϕ · d R = ϕ(Q) − ϕ(P )
F = ∇×G ⇔ G(x, y, z) =
0
F(tx, ty, tz) ×R t dt
Unit Normal:
to a surface R = R(u, v)
n =
∂uR × ∂vR
|∂uR × ∂vR|
to a surface f (x, y, z) = C
n =
∇f
|∇f |
Surface Element
dS = ∂uR × ∂vR du dv
d S = |∂uR × ∂vR| du dv
For a surface given by
z = f (x, y), a ≤ x ≤ b, y 1 (x) ≤ y ≤ y 2 (x)
dS =
1 + (∂xf )^2 + (∂yf )^2 dy dx =
dx dy
| cos γ|
Flux through S
∫ ∫
S
F · dS =
S
F · n d S
∫ (^) b
a
∫ (^) y 2 (x)
y 1 (x)
F · n
1 + (∂xf )
2
2 dy dx
Divergence Theorem
∫ ∫ ∫
D
∇ · F dV =
S
F · dS
Green’s Theorem
∫
C
(F 1 dx + F 2 dy) =
D
(∂xF 2 − ∂yF 1 )dxdy
Stokes’ Theorem
∫ ∫
D
(∇ × F ) · dS =
C
F · d R
