Differential Operators Flashcards
v(t) =
dr(t)/dt
a(t) =
dv(t)/dt = d^2 r(t)/dt^2
Continuity equation
∇ j + d/dt p = 0
Parameterise r
via u
r = (cosu,sinu,0)
cylindrical limits
0 < Φ < 2π
spherical limits
0 < Φ < 2π
0 < θ < π
cylindrical coordinates
ρ = √x^2+y^2
Φ = arctan(y/x)
z
spherical coordinates
ρ = √x^2+y^2 + z^2
Φ = arctan(y/x)
θ = arctan(√x^2+y^2 / z)
∇ · (b × c)
∇ · (b × c) =∇(b) · (b × c) + ∇(c) · (b × c)
= c · (∇(b) × b) − ∇(c) · (c × b)
= c · (∇ × b) − b · (∇ × c)
Solenoidal
∇ . a = 0
Irrotational
∇ x a = 0
to find the positions of the sources and sinks
find the divergence twice then = 0
electrostatic force
F = - ∇V
velocity or speed =
dr/dt
directional derivative
∇aΦ = (∇Φ).a
to test whether the field is conservative
take the curl if it is = 0
then it is conservative
∇ . scalar field
is a conservative field
