Categories:
Mathematics ,
Physics .
Parabolic cylindrical coordinates
Parabolic cylindrical coordinates extend parabolic coordinates ( σ , τ ) (\sigma, \tau) ( σ , τ ) ( σ , τ , z ) (\sigma, \tau, z) ( σ , τ , z ) z z z cylindrical ),
while the coordinate lines of σ \sigma σ τ \tau τ
Cartesian coordinates ( x , y , z ) (x, y, z) ( x , y , z ) ( σ , τ , z ) (\sigma, \tau, z) ( σ , τ , z )
x = 1 2 ( τ 2 − σ 2 ) y = σ τ z = z \begin{aligned}
\boxed{
\begin{aligned}
x
&= \frac{1}{2} (\tau^2 - \sigma^2)
\\
y
&= \sigma \tau
\\
z
&= z
\end{aligned}
}
\end{aligned} x y z = 2 1 ( τ 2 − σ 2 ) = σ τ = z Conversely, a point given in ( x , y , z ) (x, y, z) ( x , y , z ) ( σ , τ , z ) (\sigma, \tau, z) ( σ , τ , z )
σ = x 2 + y 2 − x τ = s g n ( y ) x 2 + y 2 + x z = z \begin{aligned}
\boxed{
\begin{aligned}
\sigma
&= \sqrt{\sqrt{x^2 + y^2} - x}
\\
\tau
&= \sgn(y) \sqrt{\sqrt{x^2 + y^2} + x}
\\
z
&= z
\end{aligned}
}
\end{aligned} σ τ z = x 2 + y 2 − x = sgn ( y ) x 2 + y 2 + x = z Parabolic cylindrical coordinates form
an orthogonal curvilinear system ,
whose scale factors h σ h_\sigma h σ h τ h_\tau h τ h z h_z h z
h σ e ^ σ = e ^ x ∂ x ∂ σ + e ^ y ∂ y ∂ σ + e ^ z ∂ z ∂ σ = − e ^ x σ + e ^ y τ h τ e ^ τ = e ^ x ∂ x ∂ τ + e ^ y ∂ y ∂ τ + e ^ z ∂ z ∂ τ = e ^ x τ + e ^ y σ h σ e ^ σ = e ^ x ∂ x ∂ z + e ^ y ∂ y ∂ z + e ^ z ∂ z ∂ z = e ^ z \begin{aligned}
h_\sigma \vu{e}_\sigma
&= \vu{e}_x \pdv{x}{\sigma} + \vu{e}_y \pdv{y}{\sigma} + \vu{e}_z \pdv{z}{\sigma}
\\
&= - \vu{e}_x \sigma + \vu{e}_y \tau
\\
h_\tau \vu{e}_\tau
&= \vu{e}_x \pdv{x}{\tau} + \vu{e}_y \pdv{y}{\tau} + \vu{e}_z \pdv{z}{\tau}
\\
&= \vu{e}_x \tau + \vu{e}_y \sigma
\\
h_\sigma \vu{e}_\sigma
&= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z}
\\
&= \vu{e}_z
\end{aligned} h σ e ^ σ h τ e ^ τ h σ e ^ σ = e ^ x ∂ σ ∂ x + e ^ y ∂ σ ∂ y + e ^ z ∂ σ ∂ z = − e ^ x σ + e ^ y τ = e ^ x ∂ τ ∂ x + e ^ y ∂ τ ∂ y + e ^ z ∂ τ ∂ z = e ^ x τ + e ^ y σ = e ^ x ∂ z ∂ x + e ^ y ∂ z ∂ y + e ^ z ∂ z ∂ z = e ^ z By normalizing the local basis vectors
e ^ σ \vu{e}_\sigma e ^ σ e ^ τ \vu{e}_\tau e ^ τ e ^ z \vu{e}_z e ^ z ρ \rho ρ
h σ = ρ ≡ σ 2 + τ 2 h τ = ρ ≡ σ 2 + τ 2 h z = 1 e ^ σ = − e ^ x σ ρ + e ^ y τ ρ e ^ τ = e ^ x τ ρ + e ^ y σ ρ e ^ z = e ^ z \begin{aligned}
\boxed{
\begin{aligned}
h_\sigma
&= \rho
\equiv \sqrt{\sigma^2 + \tau^2}
\\
h_\tau
&= \rho
\equiv \sqrt{\sigma^2 + \tau^2}
\\
h_z
&= 1
\end{aligned}
}
\qquad\qquad
\boxed{
\begin{aligned}
\vu{e}_\sigma
&= - \vu{e}_x \frac{\sigma}{\rho} + \vu{e}_y \frac{\tau}{\rho}
\\
\vu{e}_\tau
&= \vu{e}_x \frac{\tau}{\rho} + \vu{e}_y \frac{\sigma}{\rho}
\\
\vu{e}_z
&= \vu{e}_z
\end{aligned}
}
\end{aligned} h σ h τ h z = ρ ≡ σ 2 + τ 2 = ρ ≡ σ 2 + τ 2 = 1 e ^ σ e ^ τ e ^ z = − e ^ x ρ σ + e ^ y ρ τ = e ^ x ρ τ + e ^ y ρ σ = e ^ z Thanks to these scale factors, we can easily convert calculus from the Cartesian system
using the standard formulae for orthogonal curvilinear coordinates.
Differential elements
For line integrals,
the tangent vector element d ℓ \dd{\vb{\ell}} d ℓ
d ℓ = e ^ σ ρ d σ + e ^ τ ρ d τ + e ^ z d z \begin{aligned}
\boxed{
\dd{\vb{\ell}}
= \vu{e}_\sigma \: \rho \dd{\sigma} + \: \vu{e}_\tau \: \rho \dd{\tau} + \: \vu{e}_z \dd{z}
}
\end{aligned} d ℓ = e ^ σ ρ d σ + e ^ τ ρ d τ + e ^ z d z For surface integrals,
the normal vector element d S \dd{\vb{S}} d S
d S = e ^ σ ρ d τ d z + e ^ τ ρ d σ d z + e ^ z ρ 2 d σ d τ \begin{aligned}
\boxed{
\dd{\vb{S}}
= \vu{e}_\sigma \: \rho \dd{\tau} \dd{z} + \: \vu{e}_\tau \: \rho \dd{\sigma} \dd{z} + \: \vu{e}_z \: \rho^2 \dd{\sigma} \dd{\tau}
}
\end{aligned} d S = e ^ σ ρ d τ d z + e ^ τ ρ d σ d z + e ^ z ρ 2 d σ d τ And for volume integrals,
the infinitesimal volume d V \dd{V} d V
d V = ρ 2 d σ d τ d z \begin{aligned}
\boxed{
\dd{V}
= \rho^2 \dd{\sigma} \dd{\tau} \dd{z}
}
\end{aligned} d V = ρ 2 d σ d τ d z Common operations
The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
∇ f = e ^ σ 1 ρ ∂ f ∂ σ + e ^ τ 1 ρ ∂ f ∂ τ + e ^ z ∂ f ∂ z \begin{aligned}
\boxed{
\nabla f
= \vu{e}_\sigma \frac{1}{\rho} \pdv{f}{\sigma}
+ \vu{e}_\tau \frac{1}{\rho} \pdv{f}{\tau}
+ \vu{e}_z \pdv{f}{z}
}
\end{aligned} ∇ f = e ^ σ ρ 1 ∂ σ ∂ f + e ^ τ ρ 1 ∂ τ ∂ f + e ^ z ∂ z ∂ f ∇ ⋅ V = 1 ρ ∂ V σ ∂ σ + σ V σ ρ 3 + 1 ρ ∂ V τ ∂ τ + τ V τ ρ 3 + ∂ V z ∂ z \begin{aligned}
\boxed{
\nabla \cdot \vb{V}
= \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\sigma V_\sigma}{\rho^3}
+ \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\tau V_\tau}{\rho^3}
+ \pdv{V_z}{z}
}
\end{aligned} ∇ ⋅ V = ρ 1 ∂ σ ∂ V σ + ρ 3 σ V σ + ρ 1 ∂ τ ∂ V τ + ρ 3 τ V τ + ∂ z ∂ V z ∇ × V = e ^ σ ( 1 ρ ∂ V z ∂ τ − ∂ V τ ∂ z ) + e ^ τ ( ∂ V σ ∂ z − 1 ρ ∂ V z ∂ σ ) + e ^ z ( 1 ρ ∂ V τ ∂ σ + σ V τ ρ 3 − 1 ρ ∂ V σ ∂ τ − τ V σ ρ 3 ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla \times \vb{V}
&= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \bigg)
\\
&\quad\: + \vu{e}_\tau \bigg( \pdv{V_\sigma}{z} - \frac{1}{\rho} \pdv{V_z}{\sigma} \bigg)
\\
&\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} + \frac{\sigma V_\tau}{\rho^3}
- \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\tau V_\sigma}{\rho^3} \bigg)
\end{aligned}
}
\end{aligned} ∇ × V = e ^ σ ( ρ 1 ∂ τ ∂ V z − ∂ z ∂ V τ ) + e ^ τ ( ∂ z ∂ V σ − ρ 1 ∂ σ ∂ V z ) + e ^ z ( ρ 1 ∂ σ ∂ V τ + ρ 3 σ V τ − ρ 1 ∂ τ ∂ V σ − ρ 3 τ V σ ) ∇ 2 f = 1 ρ 2 ∂ 2 f ∂ σ 2 + 1 ρ 2 ∂ 2 f ∂ τ 2 + ∂ 2 f ∂ z 2 \begin{aligned}
\boxed{
\nabla^2 f
= \frac{1}{\rho^2} \pdvn{2}{f}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{f}{\tau} + \pdvn{2}{f}{z}
}
\end{aligned} ∇ 2 f = ρ 2 1 ∂ σ 2 ∂ 2 f + ρ 2 1 ∂ τ 2 ∂ 2 f + ∂ z 2 ∂ 2 f Uncommon operations
Uncommon operations include:
the gradient of a divergence ∇ ( ∇ ⋅ V ) \nabla (\nabla \cdot \vb{V}) ∇ ( ∇ ⋅ V ) ∇ V \nabla \vb{V} ∇ V ( U ⋅ ∇ ) V (\vb{U} \cdot \nabla) \vb{V} ( U ⋅ ∇ ) V U \vb{U} U ∇ 2 V \nabla^2 \vb{V} ∇ 2 V ∇ ⋅ T ‾ ‾ \nabla \cdot \overline{\overline{\vb{T}}} ∇ ⋅ T
∇ ( ∇ ⋅ V ) = e ^ σ ( 1 ρ 2 ∂ 2 V σ ∂ σ 2 + 1 ρ 2 ∂ 2 V τ ∂ σ ∂ τ + 1 ρ ∂ 2 V z ∂ σ ∂ z + τ ρ 4 ∂ V τ ∂ σ − σ ρ 4 ∂ V τ ∂ τ + ρ 2 − 3 σ 2 ρ 6 V σ − 3 σ τ V τ ρ 6 ) + e ^ τ ( 1 ρ 2 ∂ 2 V σ ∂ τ ∂ σ + 1 ρ 2 ∂ 2 V τ ∂ τ 2 + 1 ρ ∂ 2 V z ∂ τ ∂ z − τ ρ 4 ∂ V σ ∂ σ + σ ρ 4 ∂ V σ ∂ τ − 3 σ τ V σ ρ 6 + ρ 2 − 3 τ 2 ρ 6 V τ ) + e ^ z ( 1 ρ ∂ 2 V σ ∂ z ∂ σ + 1 ρ ∂ 2 V τ ∂ z ∂ τ + ∂ 2 V z ∂ z 2 + σ ρ 3 ∂ V σ ∂ z + τ ρ 3 ∂ V τ ∂ z ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla (\nabla \cdot \vb{V})
&= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \mpdv{V_\tau}{\sigma}{\tau}
+ \frac{1}{\rho} \mpdv{V_z}{\sigma}{z}
\\
&\qquad\qquad + \frac{\tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{\sigma}{\rho^4} \pdv{V_\tau}{\tau}
+ \frac{\rho^2 - 3 \sigma^2}{\rho^6} V_\sigma - \frac{3 \sigma \tau V_\tau}{\rho^6} \bigg)
\\
&\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \mpdv{V_\sigma}{\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau}
+ \frac{1}{\rho} \mpdv{V_z}{\tau}{z}
\\
&\qquad\qquad - \frac{\tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{\sigma}{\rho^4} \pdv{V_\sigma}{\tau}
- \frac{3 \sigma \tau V_\sigma}{\rho^6} + \frac{\rho^2 - 3 \tau^2}{\rho^6} V_\tau \bigg)
\\
&\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \mpdv{V_\sigma}{z}{\sigma} + \frac{1}{\rho} \mpdv{V_\tau}{z}{\tau} + \pdvn{2}{V_z}{z}
+ \frac{\sigma}{\rho^3} \pdv{V_\sigma}{z} + \frac{\tau}{\rho^3} \pdv{V_\tau}{z} \bigg)
\end{aligned}
}
\end{aligned} ∇ ( ∇ ⋅ V ) = e ^ σ ( ρ 2 1 ∂ σ 2 ∂ 2 V σ + ρ 2 1 ∂ σ ∂ τ ∂ 2 V τ + ρ 1 ∂ σ ∂ z ∂ 2 V z + ρ 4 τ ∂ σ ∂ V τ − ρ 4 σ ∂ τ ∂ V τ + ρ 6 ρ 2 − 3 σ 2 V σ − ρ 6 3 σ τ V τ ) + e ^ τ ( ρ 2 1 ∂ τ ∂ σ ∂ 2 V σ + ρ 2 1 ∂ τ 2 ∂ 2 V τ + ρ 1 ∂ τ ∂ z ∂ 2 V z − ρ 4 τ ∂ σ ∂ V σ + ρ 4 σ ∂ τ ∂ V σ − ρ 6 3 σ τ V σ + ρ 6 ρ 2 − 3 τ 2 V τ ) + e ^ z ( ρ 1 ∂ z ∂ σ ∂ 2 V σ + ρ 1 ∂ z ∂ τ ∂ 2 V τ + ∂ z 2 ∂ 2 V z + ρ 3 σ ∂ z ∂ V σ + ρ 3 τ ∂ z ∂ V τ ) ∇ V = e ^ σ e ^ σ ( 1 ρ ∂ V σ ∂ σ + τ V τ ρ 3 ) + e ^ σ e ^ τ ( 1 ρ ∂ V τ ∂ σ − τ V σ ρ 3 ) + e ^ σ e ^ z 1 ρ ∂ V z ∂ σ + e ^ τ e ^ σ ( 1 ρ ∂ V σ ∂ τ − σ V τ ρ 3 ) + e ^ τ e ^ τ ( 1 ρ ∂ V τ ∂ τ + σ V σ ρ 3 ) + e ^ τ e ^ z 1 ρ ∂ V z ∂ τ + e ^ z e ^ σ ∂ V σ ∂ z + e ^ z e ^ τ ∂ V τ ∂ z + e ^ z e ^ z ∂ V z ∂ z \begin{aligned}
\boxed{
\begin{aligned}
\nabla \vb{V}
&= \quad \vu{e}_\sigma \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\tau V_\tau}{\rho^3} \bigg)
+ \vu{e}_\sigma \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} - \frac{\tau V_\sigma}{\rho^3} \bigg)
+ \vu{e}_\sigma \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\sigma}
\\
&\quad\: + \vu{e}_\tau \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\sigma V_\tau}{\rho^3} \bigg)
+ \vu{e}_\tau \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\sigma V_\sigma}{\rho^3} \bigg)
+ \vu{e}_\tau \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\tau}
\\
&\quad\: + \vu{e}_z \vu{e}_\sigma \pdv{V_\sigma}{z}
+ \vu{e}_z \vu{e}_\tau \pdv{V_\tau}{z}
+ \vu{e}_z \vu{e}_z \pdv{V_z}{z}
\end{aligned}
}
\end{aligned} ∇ V = e ^ σ e ^ σ ( ρ 1 ∂ σ ∂ V σ + ρ 3 τ V τ ) + e ^ σ e ^ τ ( ρ 1 ∂ σ ∂ V τ − ρ 3 τ V σ ) + e ^ σ e ^ z ρ 1 ∂ σ ∂ V z + e ^ τ e ^ σ ( ρ 1 ∂ τ ∂ V σ − ρ 3 σ V τ ) + e ^ τ e ^ τ ( ρ 1 ∂ τ ∂ V τ + ρ 3 σ V σ ) + e ^ τ e ^ z ρ 1 ∂ τ ∂ V z + e ^ z e ^ σ ∂ z ∂ V σ + e ^ z e ^ τ ∂ z ∂ V τ + e ^ z e ^ z ∂ z ∂ V z ( U ⋅ ∇ ) V = e ^ σ ( U σ ρ ∂ V σ ∂ σ + U τ ρ ∂ V σ ∂ τ + U z ∂ V σ ∂ z + τ ρ 3 U σ V τ − σ ρ 3 U τ V τ ) + e ^ τ ( U σ ρ ∂ V τ ∂ σ + U τ ρ ∂ V τ ∂ τ + U z ∂ V τ ∂ z + σ ρ 3 U τ V σ − τ ρ 3 U σ V σ ) + e ^ z ( U σ ρ ∂ V z ∂ σ + U τ ρ ∂ V z ∂ τ + U z ∂ V z ∂ z ) \begin{aligned}
\boxed{
\begin{aligned}
(\vb{U} \cdot \nabla) \vb{V}
&= \quad \vu{e}_\sigma \bigg( \frac{U_\sigma}{\rho} \pdv{V_\sigma}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\sigma}{\tau} + U_z \pdv{V_\sigma}{z}
+ \frac{\tau}{\rho^3} U_\sigma V_\tau - \frac{\sigma}{\rho^3} U_\tau V_\tau \bigg)
\\
&\quad\: + \vu{e}_\tau \bigg( \frac{U_\sigma}{\rho} \pdv{V_\tau}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\tau}{\tau} + U_z \pdv{V_\tau}{z}
+ \frac{\sigma}{\rho^3} U_\tau V_\sigma - \frac{\tau}{\rho^3} U_\sigma V_\sigma \bigg)
\\
&\quad\: + \vu{e}_z \bigg( \frac{U_\sigma}{\rho} \pdv{V_z}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_z}{\tau} + U_z \pdv{V_z}{z} \bigg)
\end{aligned}
}
\end{aligned} ( U ⋅ ∇ ) V = e ^ σ ( ρ U σ ∂ σ ∂ V σ + ρ U τ ∂ τ ∂ V σ + U z ∂ z ∂ V σ + ρ 3 τ U σ V τ − ρ 3 σ U τ V τ ) + e ^ τ ( ρ U σ ∂ σ ∂ V τ + ρ U τ ∂ τ ∂ V τ + U z ∂ z ∂ V τ + ρ 3 σ U τ V σ − ρ 3 τ U σ V σ ) + e ^ z ( ρ U σ ∂ σ ∂ V z + ρ U τ ∂ τ ∂ V z + U z ∂ z ∂ V z ) ∇ 2 V = e ^ σ ( 1 ρ 2 ∂ 2 V σ ∂ σ 2 + 1 ρ 2 ∂ 2 V σ ∂ τ 2 + ∂ 2 V σ ∂ z 2 + 2 τ ρ 4 ∂ V τ ∂ σ − 2 σ ρ 4 ∂ V τ ∂ τ − V σ ρ 4 ) + e ^ τ ( 1 ρ 2 ∂ 2 V τ ∂ σ 2 + 1 ρ 2 ∂ 2 V τ ∂ τ 2 + ∂ 2 V τ ∂ z 2 − 2 τ ρ 4 ∂ V σ ∂ σ + 2 σ ρ 4 ∂ V σ ∂ τ − V τ ρ 4 ) + e ^ z ( 1 ρ 2 ∂ 2 V z ∂ σ 2 + 1 ρ 2 ∂ 2 V z ∂ τ 2 + ∂ 2 V z ∂ z 2 ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla^2 \vb{V}
&= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\tau} + \pdvn{2}{V_\sigma}{z}
+ \frac{2 \tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{2 \sigma}{\rho^4} \pdv{V_\tau}{\tau} - \frac{V_\sigma}{\rho^4} \bigg)
\\
&\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + \pdvn{2}{V_\tau}{z}
- \frac{2 \tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{2 \sigma}{\rho^4} \pdv{V_\sigma}{\tau} - \frac{V_\tau}{\rho^4} \bigg)
\\
&\quad\: + \vu{e}_z \bigg( \frac{1}{\rho^2} \pdvn{2}{V_z}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_z}{\tau} + \pdvn{2}{V_z}{z} \bigg)
\end{aligned}
}
\end{aligned} ∇ 2 V = e ^ σ ( ρ 2 1 ∂ σ 2 ∂ 2 V σ + ρ 2 1 ∂ τ 2 ∂ 2 V σ + ∂ z 2 ∂ 2 V σ + ρ 4 2 τ ∂ σ ∂ V τ − ρ 4 2 σ ∂ τ ∂ V τ − ρ 4 V σ ) + e ^ τ ( ρ 2 1 ∂ σ 2 ∂ 2 V τ + ρ 2 1 ∂ τ 2 ∂ 2 V τ + ∂ z 2 ∂ 2 V τ − ρ 4 2 τ ∂ σ ∂ V σ + ρ 4 2 σ ∂ τ ∂ V σ − ρ 4 V τ ) + e ^ z ( ρ 2 1 ∂ σ 2 ∂ 2 V z + ρ 2 1 ∂ τ 2 ∂ 2 V z + ∂ z 2 ∂ 2 V z ) ∇ ⋅ T ‾ ‾ = e ^ σ ( 1 ρ ∂ T σ σ ∂ σ + 1 ρ ∂ T τ σ ∂ τ + ∂ T z σ ∂ z + σ T σ σ ρ 3 + τ T σ τ ρ 3 + τ T τ σ ρ 3 − σ T τ τ ρ 3 ) + e ^ τ ( 1 ρ ∂ T σ τ ∂ σ + 1 ρ ∂ T τ τ ∂ τ + ∂ T k τ ∂ z − τ T σ σ ρ 3 + σ T σ τ ρ 3 + σ T τ σ ρ 3 + τ T τ τ ρ 3 ) + e ^ z ( 1 ρ ∂ T σ z ∂ σ + 1 ρ ∂ T τ z ∂ τ + ∂ T z z ∂ z + σ T σ z ρ 3 + τ T τ z ρ 3 ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla \cdot \overline{\overline{\mathbf{T}}}
&= \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{T_{\sigma \sigma}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \sigma}}{\tau} + \pdv{T_{z \sigma}}{z}
+ \frac{\sigma T_{\sigma \sigma}}{\rho^3} + \frac{\tau T_{\sigma \tau}}{\rho^3}
+ \frac{\tau T_{\tau \sigma}}{\rho^3} - \frac{\sigma T_{\tau \tau}}{\rho^3} \bigg)
\\
&+ \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{T_{\sigma \tau}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \tau}}{\tau} + \pdv{T_{k \tau}}{z}
- \frac{\tau T_{\sigma \sigma}}{\rho^3} + \frac{\sigma T_{\sigma \tau}}{\rho^3}
+ \frac{\sigma T_{\tau \sigma}}{\rho^3} + \frac{\tau T_{\tau \tau}}{\rho^3} \bigg)
\\
&+ \vu{e}_z \bigg( \frac{1}{\rho} \pdv{T_{\sigma z}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau z}}{\tau} + \pdv{T_{zz}}{z}
+ \frac{\sigma T_{\sigma z}}{\rho^3} + \frac{\tau T_{\tau z}}{\rho^3} \bigg)
\end{aligned}
}
\end{aligned} ∇ ⋅ T = e ^ σ ( ρ 1 ∂ σ ∂ T σ σ + ρ 1 ∂ τ ∂ T τ σ + ∂ z ∂ T z σ + ρ 3 σ T σ σ + ρ 3 τ T σ τ + ρ 3 τ T τ σ − ρ 3 σ T τ τ ) + e ^ τ ( ρ 1 ∂ σ ∂ T σ τ + ρ 1 ∂ τ ∂ T τ τ + ∂ z ∂ T k τ − ρ 3 τ T σ σ + ρ 3 σ T σ τ + ρ 3 σ T τ σ + ρ 3 τ T τ τ ) + e ^ z ( ρ 1 ∂ σ ∂ T σ z + ρ 1 ∂ τ ∂ T τ z + ∂ z ∂ T z z + ρ 3 σ T σ z + ρ 3 τ T τ z ) References
M.L. Boas,
Mathematical methods in the physical sciences , 2nd edition,
Wiley.