Categories:
Mathematics ,
Physics .
Spherical coordinates
Spherical coordinates are an extension of polar coordinates ( r , φ ) (r, \varphi) ( r , φ ) ( r , θ , φ ) (r, \theta, \varphi) ( r , θ , φ )
r r r radius or radial distance : distance to the origin.θ \theta θ elevation , polar angle or colatitude :
angle to the positive z z z zenith , i.e. the “north pole”.φ \varphi φ azimuth , azimuthal angle or longitude :
angle from the positive x x x
Note that this is the standard notation among physicists,
but mathematicians often switch the definitions of θ \theta θ φ \varphi φ ( r , θ , φ ) (r, \theta, \varphi) ( r , θ , φ )
Cartesian coordinates ( x , y , z ) (x, y, z) ( x , y , z ) ( r , θ , φ ) (r, \theta, \varphi) ( r , θ , φ )
x = r sin θ cos φ y = r sin θ sin φ z = r cos θ \begin{aligned}
\boxed{
\begin{aligned}
x &= r \sin\theta \cos\varphi \\
y &= r \sin\theta \sin\varphi \\
z &= r \cos\theta
\end{aligned}
}
\end{aligned} x y z = r sin θ cos φ = r sin θ sin φ = r cos θ Conversely, a point given in ( x , y , z ) (x, y, z) ( x , y , z ) ( r , θ , φ ) (r, \theta, \varphi) ( r , θ , φ ) a t a n 2 \mathtt{atan2} atan2
r = x 2 + y 2 + z 2 θ = arccos ( z / r ) φ = a t a n 2 ( y , x ) \begin{aligned}
\boxed{
\begin{aligned}
r
&= \sqrt{x^2 + y^2 + z^2}
\\
\theta
&= \arccos(z / r)
\\
\varphi
&= \mathtt{atan2}(y, x)
\end{aligned}
}
\end{aligned} r θ φ = x 2 + y 2 + z 2 = arccos ( z / r ) = atan2 ( y , x ) Spherical coordinates form
an orthogonal curvilinear system ,
whose scale factors h r h_r h r h θ h_\theta h θ h φ h_\varphi h φ
h r e ^ r = e ^ x ∂ x ∂ r + e ^ y ∂ y ∂ r + e ^ z ∂ z ∂ r = e ^ x sin θ cos φ + e ^ y sin θ sin φ + e ^ z cos θ h θ e ^ θ = e ^ x ∂ x ∂ θ + e ^ y ∂ y ∂ θ + e ^ z ∂ z ∂ θ = e ^ x r cos θ cos φ + e ^ y r cos θ sin φ − e ^ z r sin θ h φ e ^ φ = e ^ x ∂ x ∂ φ + e ^ y ∂ y ∂ φ + e ^ z ∂ z ∂ φ = − e ^ x r sin θ sin φ + e ^ y r sin θ cos φ \begin{aligned}
h_r \vu{e}_r
&= \vu{e}_x \pdv{x}{r} + \vu{e}_y \pdv{y}{r} + \vu{e}_z \pdv{z}{r}
\\
&= \vu{e}_x \sin{\theta} \cos{\varphi} + \vu{e}_y \sin{\theta} \sin{\varphi} + \vu{e}_z \cos{\theta}
\\
h_\theta \vu{e}_\theta
&= \vu{e}_x \pdv{x}{\theta} + \vu{e}_y \pdv{y}{\theta} + \vu{e}_z \pdv{z}{\theta}
\\
&= \vu{e}_x \: r \cos{\theta} \cos{\varphi} + \vu{e}_y \: r \cos{\theta} \sin{\varphi} - \vu{e}_z \: r \sin{\theta}
\\
h_\varphi \vu{e}_\varphi
&= \vu{e}_x \pdv{x}{\varphi} + \vu{e}_y \pdv{y}{\varphi} + \vu{e}_z \pdv{z}{\varphi}
\\
&= - \vu{e}_x \: r \sin{\theta} \sin{\varphi} + \vu{e}_y \: r \sin{\theta} \cos{\varphi}
\end{aligned} h r e ^ r h θ e ^ θ h φ e ^ φ = e ^ x ∂ r ∂ x + e ^ y ∂ r ∂ y + e ^ z ∂ r ∂ z = e ^ x sin θ cos φ + e ^ y sin θ sin φ + e ^ z cos θ = e ^ x ∂ θ ∂ x + e ^ y ∂ θ ∂ y + e ^ z ∂ θ ∂ z = e ^ x r cos θ cos φ + e ^ y r cos θ sin φ − e ^ z r sin θ = e ^ x ∂ φ ∂ x + e ^ y ∂ φ ∂ y + e ^ z ∂ φ ∂ z = − e ^ x r sin θ sin φ + e ^ y r sin θ cos φ By normalizing the local basis vectors
e ^ r \vu{e}_r e ^ r e ^ θ \vu{e}_\theta e ^ θ e ^ φ \vu{e}_\varphi e ^ φ
h r = 1 h θ = r h φ = r sin θ e ^ r = e ^ x sin θ cos φ + e ^ y sin θ sin φ + e ^ z cos θ e ^ θ = e ^ x cos θ cos φ + e ^ y cos θ sin φ − e ^ z sin θ e ^ φ = − e ^ x sin φ + e ^ y cos φ \begin{aligned}
\boxed{
\begin{aligned}
h_r
&= 1
\\
h_\theta
&= r
\\
h_\varphi
&= r \sin{\theta}
\end{aligned}
}
\qquad\qquad
\boxed{
\begin{aligned}
\vu{e}_r
&= \vu{e}_x \sin{\theta} \cos{\varphi} + \vu{e}_y \sin{\theta} \sin{\varphi} + \vu{e}_z \cos{\theta}
\\
\vu{e}_\theta
&= \vu{e}_x \cos{\theta} \cos{\varphi} + \vu{e}_y \cos{\theta} \sin{\varphi} - \vu{e}_z \sin{\theta}
\\
\vu{e}_\varphi
&= - \vu{e}_x \sin{\varphi} + \vu{e}_y \cos{\varphi}
\end{aligned}
}
\end{aligned} h r h θ h φ = 1 = r = r sin θ e ^ r e ^ θ e ^ φ = e ^ x sin θ cos φ + e ^ y sin θ sin φ + e ^ z cos θ = e ^ x cos θ cos φ + e ^ y cos θ sin φ − e ^ z sin θ = − e ^ x sin φ + e ^ y cos φ 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 ^ r d r + e ^ θ r d θ + e ^ φ r sin θ d φ \begin{aligned}
\boxed{
\dd{\vb{\ell}}
= \vu{e}_r \dd{r}
+ \: \vu{e}_\theta \: r \dd{\theta}
+ \: \vu{e}_\varphi \: r \sin{\theta} \dd{\varphi}
}
\end{aligned} d ℓ = e ^ r d r + e ^ θ r d θ + e ^ φ r sin θ d φ For surface integrals,
the normal vector element d S \dd{\vb{S}} d S
d S = e ^ r r 2 sin θ d θ d φ + e ^ θ r sin θ d r d φ + e ^ φ r d r d θ \begin{aligned}
\boxed{
\dd{\vb{S}}
= \vu{e}_r \: r^2 \sin{\theta} \dd{\theta} \dd{\varphi}
+ \: \vu{e}_\theta \: r \sin{\theta} \dd{r} \dd{\varphi}
+ \: \vu{e}_\varphi \: r \dd{r} \dd{\theta}
}
\end{aligned} d S = e ^ r r 2 sin θ d θ d φ + e ^ θ r sin θ d r d φ + e ^ φ r d r d θ And for volume integrals,
the infinitesimal volume d V \dd{V} d V
d V = r 2 sin θ d r d θ d φ \begin{aligned}
\boxed{
\dd{V}
= r^2 \sin{\theta} \dd{r} \dd{\theta} \dd{\varphi}
}
\end{aligned} d V = r 2 sin θ d r d θ d φ Common operations
The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
∇ f = e ^ r ∂ f ∂ r + e ^ θ 1 r ∂ f ∂ θ + e φ 1 r sin θ ∂ f ∂ φ \begin{aligned}
\boxed{
\nabla f
= \vu{e}_r \pdv{f}{r}
+ \vu{e}_\theta \frac{1}{r} \pdv{f}{\theta} + \mathbf{e}_\varphi \frac{1}{r \sin{\theta}} \pdv{f}{\varphi}
}
\end{aligned} ∇ f = e ^ r ∂ r ∂ f + e ^ θ r 1 ∂ θ ∂ f + e φ r sin θ 1 ∂ φ ∂ f ∇ ⋅ V = ∂ V r ∂ r + 2 V r r + 1 r ∂ V θ ∂ θ + V θ r tan θ + 1 r sin θ ∂ V φ ∂ φ \begin{aligned}
\boxed{
\nabla \cdot \vb{V}
= \pdv{V_r}{r} + \frac{2 V_r}{r}
+ \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{V_\theta}{r \tan{\theta}}
+ \frac{1}{r \sin\theta} \pdv{V_\varphi}{\varphi}
}
\end{aligned} ∇ ⋅ V = ∂ r ∂ V r + r 2 V r + r 1 ∂ θ ∂ V θ + r tan θ V θ + r sin θ 1 ∂ φ ∂ V φ ∇ × V = e ^ r ( 1 r ∂ V φ ∂ θ + V φ r tan θ − 1 r sin θ ∂ V θ ∂ φ ) + e ^ θ ( 1 r sin θ ∂ V r ∂ φ − ∂ V φ ∂ r − V φ r ) + e ^ φ ( ∂ V θ ∂ r + V θ r − 1 r ∂ V r ∂ θ ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla \times \vb{V}
&= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_\varphi}{\theta} + \frac{V_\varphi}{r \tan{\theta}}
- \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} \bigg)
\\
&\quad\: + \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi}
- \pdv{V_\varphi}{r} - \frac{V_\varphi}{r} \bigg)
\\
&\quad\: + \vu{e}_\varphi \bigg( \pdv{V_\theta}{r} + \frac{V_\theta}{r}
- \frac{1}{r} \pdv{V_r}{\theta} \bigg)
\end{aligned}
}
\end{aligned} ∇ × V = e ^ r ( r 1 ∂ θ ∂ V φ + r tan θ V φ − r sin θ 1 ∂ φ ∂ V θ ) + e ^ θ ( r sin θ 1 ∂ φ ∂ V r − ∂ r ∂ V φ − r V φ ) + e ^ φ ( ∂ r ∂ V θ + r V θ − r 1 ∂ θ ∂ V r ) ∇ 2 f = ∂ 2 f ∂ r 2 + 2 r ∂ f ∂ r + 1 r 2 ∂ 2 f ∂ θ 2 + 1 r 2 tan θ ∂ f ∂ θ + 1 r 2 sin 2 θ ∂ 2 f ∂ φ 2 \begin{aligned}
\boxed{
\nabla^2 f
= \pdvn{2}{f}{r} + \frac{2}{r} \pdv{f}{r}
+ \frac{1}{r^2} \pdvn{2}{f}{\theta} + \frac{1}{r^2 \tan{\theta}} \pdv{f}{\theta}
+ \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{f}{\varphi}
}
\end{aligned} ∇ 2 f = ∂ r 2 ∂ 2 f + r 2 ∂ r ∂ f + r 2 1 ∂ θ 2 ∂ 2 f + r 2 tan θ 1 ∂ θ ∂ f + r 2 sin 2 θ 1 ∂ φ 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 ^ r ( ∂ 2 V r ∂ r 2 + 1 r ∂ 2 V θ ∂ r ∂ θ + 1 r sin θ ∂ 2 V φ ∂ φ ∂ r + 2 r ∂ V r ∂ r − 1 r 2 ∂ V θ ∂ θ − 1 r 2 sin θ ∂ V φ ∂ φ + 1 r tan θ ∂ V θ ∂ r − 2 V r r 2 − V θ r 2 tan θ ) + e ^ θ ( 1 r ∂ 2 V r ∂ θ ∂ r + 1 r 2 ∂ 2 V θ ∂ θ 2 + 1 r 2 sin θ ∂ 2 V φ ∂ θ ∂ φ + 2 r 2 ∂ V r ∂ θ + 1 r 2 tan θ ∂ V θ ∂ θ − cos θ r 2 sin 2 θ ∂ V φ ∂ φ − V θ r 2 sin 2 θ ) + e ^ φ ( 1 r sin θ ∂ 2 V r ∂ φ ∂ r + 1 r 2 sin θ ∂ 2 V θ ∂ φ ∂ θ + 1 r 2 sin 2 θ ∂ 2 V φ ∂ φ 2 + 2 r 2 sin θ ∂ V r ∂ φ + cos θ r 2 sin 2 θ ∂ V θ ∂ φ ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla (\nabla \cdot \vb{V})
&= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\theta}{r}{\theta} + \frac{1}{r \sin{\theta}} \mpdv{V_\varphi}{\varphi}{r}
\\
&\qquad\qquad + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta}
- \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
+ \frac{1}{r \tan{\theta}} \pdv{V_\theta}{r} - \frac{2 V_r}{r^2} - \frac{V_\theta}{r^2 \tan{\theta}} \bigg)
\\
&\quad\: + \vu{e}_\theta \bigg( \frac{1}{r} \mpdv{V_r}{\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
+ \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi}
\\
&\qquad\qquad + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
- \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
\\
&\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r \sin{\theta}} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\theta}{\varphi}{\theta}
+ \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi}
\\
&\qquad\qquad + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi} + \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi} \bigg)
\end{aligned}
}
\end{aligned} ∇ ( ∇ ⋅ V ) = e ^ r ( ∂ r 2 ∂ 2 V r + r 1 ∂ r ∂ θ ∂ 2 V θ + r sin θ 1 ∂ φ ∂ r ∂ 2 V φ + r 2 ∂ r ∂ V r − r 2 1 ∂ θ ∂ V θ − r 2 sin θ 1 ∂ φ ∂ V φ + r tan θ 1 ∂ r ∂ V θ − r 2 2 V r − r 2 tan θ V θ ) + e ^ θ ( r 1 ∂ θ ∂ r ∂ 2 V r + r 2 1 ∂ θ 2 ∂ 2 V θ + r 2 sin θ 1 ∂ θ ∂ φ ∂ 2 V φ + r 2 2 ∂ θ ∂ V r + r 2 tan θ 1 ∂ θ ∂ V θ − r 2 sin 2 θ cos θ ∂ φ ∂ V φ − r 2 sin 2 θ V θ ) + e ^ φ ( r sin θ 1 ∂ φ ∂ r ∂ 2 V r + r 2 sin θ 1 ∂ φ ∂ θ ∂ 2 V θ + r 2 sin 2 θ 1 ∂ φ 2 ∂ 2 V φ + r 2 sin θ 2 ∂ φ ∂ V r + r 2 sin 2 θ cos θ ∂ φ ∂ V θ ) ∇ V = e ^ r e ^ r ∂ V r ∂ r + e ^ r e ^ θ ∂ V θ ∂ r + e ^ r e ^ φ ∂ V φ ∂ r + e ^ θ e ^ r ( 1 r ∂ V r ∂ θ − V θ r ) + e ^ θ e ^ θ ( 1 r ∂ V θ ∂ θ + V r r ) + e ^ θ e ^ φ 1 r ∂ V φ ∂ θ + e ^ φ e ^ r ( 1 r sin θ ∂ V r ∂ φ − V φ r ) + e ^ φ e ^ θ ( 1 r sin θ ∂ V θ ∂ φ − V φ r tan θ ) + e ^ φ e ^ φ ( 1 r sin θ ∂ V φ ∂ φ + V r r + V θ r tan θ ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla \vb{V}
&= \quad \vu{e}_r \vu{e}_r \pdv{V_r}{r} + \vu{e}_r \vu{e}_\theta \pdv{V_\theta}{r} + \vu{e}_r \vu{e}_\varphi \pdv{V_\varphi}{r}
\\
&\quad\: + \vu{e}_\theta \vu{e}_r \bigg( \frac{1}{r} \pdv{V_r}{\theta} - \frac{V_\theta}{r} \bigg)
+ \vu{e}_\theta \vu{e}_\theta \bigg( \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{V_r}{r} \bigg)
+ \vu{e}_\theta \vu{e}_\varphi \frac{1}{r} \pdv{V_\varphi}{\theta}
\\
&\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg)
+ \vu{e}_\varphi \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} - \frac{V_\varphi}{r \tan{\theta}} \bigg)
\\
&\quad\: + \vu{e}_\varphi \vu{e}_\varphi
\bigg( \frac{1}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} + \frac{V_\theta}{r \tan{\theta}} \bigg)
\end{aligned}
}
\end{aligned} ∇ V = e ^ r e ^ r ∂ r ∂ V r + e ^ r e ^ θ ∂ r ∂ V θ + e ^ r e ^ φ ∂ r ∂ V φ + e ^ θ e ^ r ( r 1 ∂ θ ∂ V r − r V θ ) + e ^ θ e ^ θ ( r 1 ∂ θ ∂ V θ + r V r ) + e ^ θ e ^ φ r 1 ∂ θ ∂ V φ + e ^ φ e ^ r ( r sin θ 1 ∂ φ ∂ V r − r V φ ) + e ^ φ e ^ θ ( r sin θ 1 ∂ φ ∂ V θ − r tan θ V φ ) + e ^ φ e ^ φ ( r sin θ 1 ∂ φ ∂ V φ + r V r + r tan θ V θ ) ( U ⋅ ∇ ) V = e ^ r ( U r ∂ V r ∂ r + U θ r ∂ V r ∂ θ + U φ r sin θ ∂ V r ∂ φ − U θ V θ r − U φ V φ r ) + e ^ θ ( U r ∂ V θ ∂ r + U θ r ∂ V θ ∂ θ + U φ r sin θ ∂ V θ ∂ φ + U θ V r r − U φ V φ r tan θ ) + e ^ φ ( U r ∂ V φ ∂ r + U θ r ∂ V φ ∂ θ + U φ r sin θ ∂ V φ ∂ φ + U φ V r r + U φ V θ r tan θ ) \begin{aligned}
\boxed{
\begin{aligned}
(\vb{U} \cdot \nabla) \vb{V}
&= \quad \vu{e}_r \bigg( U_r \pdv{V_r}{r} + \frac{U_\theta}{r} \pdv{V_r}{\theta}
+ \frac{U_\varphi}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{U_\theta V_\theta}{r} - \frac{U_\varphi V_\varphi}{r} \bigg)
\\
&\quad\: + \vu{e}_\theta \bigg( U_r \pdv{V_\theta}{r} + \frac{U_\theta}{r} \pdv{V_\theta}{\theta}
+ \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\theta}{\varphi} + \frac{U_\theta V_r}{r} - \frac{U_\varphi V_\varphi}{r \tan{\theta}} \bigg)
\\
&\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\theta}{r} \pdv{V_\varphi}{\theta}
+ \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{U_\varphi V_r}{r} + \frac{U_\varphi V_\theta}{r \tan{\theta}} \bigg)
\end{aligned}
}
\end{aligned} ( U ⋅ ∇ ) V = e ^ r ( U r ∂ r ∂ V r + r U θ ∂ θ ∂ V r + r sin θ U φ ∂ φ ∂ V r − r U θ V θ − r U φ V φ ) + e ^ θ ( U r ∂ r ∂ V θ + r U θ ∂ θ ∂ V θ + r sin θ U φ ∂ φ ∂ V θ + r U θ V r − r tan θ U φ V φ ) + e ^ φ ( U r ∂ r ∂ V φ + r U θ ∂ θ ∂ V φ + r sin θ U φ ∂ φ ∂ V φ + r U φ V r + r tan θ U φ V θ ) ∇ 2 V = e ^ r ( ∂ 2 V r ∂ r 2 + 1 r 2 ∂ 2 V r ∂ θ 2 + 1 r 2 sin 2 θ ∂ 2 V r ∂ φ 2 + 2 r ∂ V r ∂ r + 1 r 2 tan θ ∂ V r ∂ θ − 2 r 2 ∂ V θ ∂ θ − 2 r 2 sin θ ∂ V φ ∂ φ − 2 V r r 2 − 2 V θ r 2 tan θ ) + e ^ θ ( ∂ 2 V θ ∂ r 2 + 1 r 2 ∂ 2 V θ ∂ θ 2 + 1 r 2 sin 2 θ ∂ 2 V θ ∂ φ 2 + 2 r 2 ∂ V r ∂ θ + 2 r ∂ V θ ∂ r + 1 r 2 tan θ ∂ V θ ∂ θ − 2 cos θ r 2 sin 2 θ ∂ V φ ∂ φ − V θ r 2 sin 2 θ ) + e ^ φ ( ∂ 2 V φ ∂ r 2 + 1 r 2 ∂ 2 V φ ∂ θ 2 + 1 r 2 sin 2 θ ∂ 2 V φ ∂ φ 2 + 2 r 2 sin θ ∂ V r ∂ φ + 2 cos θ r 2 sin 2 θ ∂ V θ ∂ φ + 2 r ∂ V φ ∂ r + 1 r 2 tan θ ∂ V φ ∂ θ − V φ r 2 sin 2 θ ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla^2 \vb{V}
&= \quad\: \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_r}{\varphi}
\\
&\qquad\qquad + \frac{2}{r} \pdv{V_r}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_r}{\theta}
- \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
- \frac{2 V_r}{r^2} - \frac{2 V_\theta}{r^2 \tan{\theta}} \bigg)
\\
&\quad\: + \vu{e}_\theta \bigg( \pdvn{2}{V_\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
+ \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi}
\\
&\qquad\qquad + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
- \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
\\
&\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\theta}
+ \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi}
\\
&\qquad\qquad + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi} + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi}
+ \frac{2}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\varphi}{\theta} - \frac{V_\varphi}{r^2 \sin^2{\theta}} \bigg)
\end{aligned}
}
\end{aligned} ∇ 2 V = e ^ r ( ∂ r 2 ∂ 2 V r + r 2 1 ∂ θ 2 ∂ 2 V r + r 2 sin 2 θ 1 ∂ φ 2 ∂ 2 V r + r 2 ∂ r ∂ V r + r 2 tan θ 1 ∂ θ ∂ V r − r 2 2 ∂ θ ∂ V θ − r 2 sin θ 2 ∂ φ ∂ V φ − r 2 2 V r − r 2 tan θ 2 V θ ) + e ^ θ ( ∂ r 2 ∂ 2 V θ + r 2 1 ∂ θ 2 ∂ 2 V θ + r 2 sin 2 θ 1 ∂ φ 2 ∂ 2 V θ + r 2 2 ∂ θ ∂ V r + r 2 ∂ r ∂ V θ + r 2 tan θ 1 ∂ θ ∂ V θ − r 2 sin 2 θ 2 cos θ ∂ φ ∂ V φ − r 2 sin 2 θ V θ ) + e ^ φ ( ∂ r 2 ∂ 2 V φ + r 2 1 ∂ θ 2 ∂ 2 V φ + r 2 sin 2 θ 1 ∂ φ 2 ∂ 2 V φ + r 2 sin θ 2 ∂ φ ∂ V r + r 2 sin 2 θ 2 cos θ ∂ φ ∂ V θ + r 2 ∂ r ∂ V φ + r 2 tan θ 1 ∂ θ ∂ V φ − r 2 sin 2 θ V φ ) ∇ ⋅ T ‾ ‾ = e ^ r ( ∂ T r r ∂ r + 1 r ∂ T θ r ∂ θ + 1 r sin θ ∂ T φ r ∂ φ + 2 T r r r + T θ r r tan θ − T θ θ r − T φ φ r ) + e ^ θ ( ∂ T r θ ∂ r + 1 r ∂ T θ θ ∂ θ + 1 r sin θ ∂ T φ θ ∂ φ + 2 T r θ r + T θ r r + T θ θ r tan θ − T φ φ r tan θ ) + e ^ φ ( ∂ T r φ ∂ r + 1 r ∂ T θ φ ∂ θ + 1 r sin θ ∂ T φ φ ∂ φ + 2 T r φ r + T θ φ r tan θ + T φ r r + T φ θ r tan θ ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla \cdot \overline{\overline{\mathbf{T}}}
&= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\theta r}}{\theta} + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi r}}{\varphi}
\\
&\qquad\qquad + \frac{2 T_{rr}}{r} + \frac{T_{\theta r}}{r \tan{\theta}}
- \frac{T_{\theta \theta}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
\\
&\quad\: + \vu{e}_\theta \bigg(\pdv{T_{r \theta}}{r} + \frac{1}{r} \pdv{T_{\theta \theta}}{\theta}
+ \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \theta}}{\varphi}
\\
&\qquad\qquad + \frac{2 T_{r \theta}}{r} + \frac{T_{\theta r}}{r}
+ \frac{T_{\theta \theta}}{r \tan{\theta}} - \frac{T_{\varphi \varphi}}{r \tan{\theta}} \bigg)
\\
&\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\theta \varphi}}{\theta}
+ \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \varphi}}{\varphi}
\\
&\qquad\qquad + \frac{2 T_{r \varphi}}{r} + \frac{T_{\theta \varphi}}{r \tan{\theta}}
+ \frac{T_{\varphi r}}{r} + \frac{T_{\varphi \theta}}{r \tan{\theta}} \bigg)
\end{aligned}
}
\end{aligned} ∇ ⋅ T = e ^ r ( ∂ r ∂ T r r + r 1 ∂ θ ∂ T θ r + r sin θ 1 ∂ φ ∂ T φ r + r 2 T r r + r tan θ T θ r − r T θ θ − r T φφ ) + e ^ θ ( ∂ r ∂ T r θ + r 1 ∂ θ ∂ T θ θ + r sin θ 1 ∂ φ ∂ T φθ + r 2 T r θ + r T θ r + r tan θ T θ θ − r tan θ T φφ ) + e ^ φ ( ∂ r ∂ T r φ + r 1 ∂ θ ∂ T θ φ + r sin θ 1 ∂ φ ∂ T φφ + r 2 T r φ + r tan θ T θ φ + r T φ r + r tan θ T φθ ) References
M.L. Boas,
Mathematical methods in the physical sciences , 2nd edition,
Wiley.
B. Lautrup,
Physics of continuous matter: exotic and everyday phenomena in the macroscopic world , 2nd edition,
CRC Press.