v =dr
dt=ρdˆeρ
dt+dρ
dtˆeρ=ρ(
dθ
dtˆeθ+ sin θdφ
dtˆeφ)
+dρ
dtˆeρ= ̇ρˆeρ+ρθ ̇ˆeθ+ρφ ̇sin θˆeφ(9 .38)Here vρ= ̇ρis the radial component of the velocity ,vθ=ρθ ̇is the polar component of
velocity and vφ=ρφ ̇sin θis the azimuthal component of velocity.
Differentiating the velocity with respect to time gives the acceleration vector
a =dv
dt=d(^2) r
dt^2
=d
dt
(
ρ ̇ˆeρ+ρθ ̇ˆeθ+ρφ ̇sin θˆeφ
)
= ̇ρd
ˆeρ
dt
- ̈ρeˆρ+ (ρθ ̇)d
ˆeθ
dt
+d
dt
(ρθ ̇)ˆeθ+ (ρφ ̇sin θ)d
eˆφ
dt
+d
dt
(ρφ ̇sin θ)ˆeφ
(9 .38)
Substitute the derivatives from equation (9.36) into the equation (9.38) and simplify
the results to show the acceleration vector in spherical coordinates is represented
a =dv
dt=d(^2) r
dt^2
=( ̈ρ−ρ(θ ̇)^2 −ρ(φ ̇)^2 sin^2 θ)ˆeρ
- (ρθ ̈+ 2 ̇ρθ ̇−ρ(φ ̇)^2 sin θcos θ)ˆeθ
- (ρφ ̈sin θ+ 2 ̇ρφ ̇sin θ+ 2ρθ ̇φ ̇cos θ)eˆφ
(9 .39)
where ̇= dtd and ̈= dtd^22 is the dot notation for the first and second time derivatives.
In spherical coordinates an element of volume is given by dV =r^2 sin θ dr dθ dφ
Introduction to Potential Theory
In this section some properties of irrotational and/or solenoidal vector fields are
derived. Recall that a vector field F=F(x, y, z )which is continuous and differentiable
in a region Ris called irrotational if curl F =∇× F = 0 at all points of Rand it is
called solenoidal if div F=∇·F = 0 at all points of R.
Some properties of irrotational vector fields are now considered. If a vector field
F is an irrotational vector field, then ∇× F = 0 and under these conditions the
vector field F is derivable from a scalar field φ=φ(x, y, z) and can be calculated by
the operation^4
F=F(x, y, z ) = F 1 (x, y, z)ˆe 1 +F 2 (x, y, z )ˆe 2 +F 3 (x, y, z )ˆe 3 =∇φ= grad φ=∂φ
∂x
ˆe 1 +∂φ
∂yeˆ 2 +∂φ
∂zˆe 3Note that it you have a choice to solve for three quantities
F 1 (x, y, z ), F 2 (x, y, z ), F 3 (x, y, z )(^4) Sometimes F =−grad φ. The selection of either a + or - sign in front of the gradient depends upon how
the vector field is being used.