Following 2009 Trangenstein "Numeric Simulations of Hyperbolic Conservation Laws", section 7.4:
Notice that his book coincides with anholonomic orthonormal basis.
So that's what I'm using here. $u_1 ... u_n$ are my coordinate chart parameters. Indexes are in anholonomic non-coordinates.
$\bar{j} =$ the coordinate basis index, ${e^\bar{j}}_j =$ change of basis from coordinate (non-Cartesian) to non-coordinate basis.
Conservation of our field F is a $\pmatrix{ p+1 \\ q}$ in space and time:
$\nabla \cdot F = 0$
with indexes:
$\nabla_\mu \cdot {F^{\mu \bar{i}_1 ... \bar{i}_p}}_{\bar{k}_1 ... \bar{k}_q} = 0$
separating time:
$
\nabla_t \cdot {F^{t \bar{i}_1 ... \bar{i}_p}}_{\bar{k}_1 ... \bar{k}_q}
+ \nabla_\bar{j} \cdot {F^{\bar{j} \bar{i}_1 ... \bar{i}_p}}_{\bar{k}_1 ... \bar{k}_q}
= 0$
assume all ${\Gamma^t}_{\alpha\beta} = 0$ and all ${\Gamma^\alpha}_{t\mu} = 0$
$
\partial_t \cdot {F^{t \bar{i}_1 ... \bar{i}_p}}_{\bar{k}_1 ... \bar{k}_q}
+ \nabla_\bar{j} \cdot {F^{\bar{j} \bar{i}_1 ... \bar{i}_p}}_{\bar{k}_1 ... \bar{k}_q}
= 0$
Let our state ${U^{\bar{i}_1 ... \bar{i}_p}}_{\bar{k}_1 ... \bar{k}_q} = {F^{t \bar{i}_1 ... \bar{i}_p}}_{\bar{k}_1 ... \bar{k}_q}$ be whatever is travelling through time:
$
\partial_t \cdot {U^{\bar{i}_1 ... \bar{i}_p}}_{\bar{k}_1 ... \bar{k}_q}
+ \nabla_{\bar{j}} \cdot {F^{\bar{j} \bar{i}_1 ... \bar{i}_p}}_{\bar{k}_1 ... \bar{k}_q}
= 0$
We will only consider $U$ to be a single $\left( \begin{matrix} p \\ q \end{matrix} \right)$-form. You can repeat for collections of tensors.
The indexes $\bar{i}_1 ... \bar{i}_p, \bar{j}_1 ... \bar{j}_q$ span our chart coordinates $u_1 ... u_n$.
Include the change-of-coordinates:
$
\partial_t \cdot (
{U^{i_1 ... i_p}}_{k_1 ... k_q}
\cdot {e^{\bar{i}_1}}_{i_1} \cdot ... \cdot {e^{\bar{i}_p}}_{i_p}
\cdot {e_{\bar{k}_1}}^{k_1} \cdot ... \cdot {e_{\bar{k}_q}}^{k_p}
)
+ \nabla_{\bar{j}} \cdot (
{F^{\bar{j} i_1 ... i_p}}_{k_1 ... k_q}
\cdot {e^{\bar{i}_1}}_{i_1} \cdot ... \cdot {e^{\bar{i}_p}}_{i_p}
\cdot {e_{\bar{k}_1}}^{k_1} \cdot ... \cdot {e_{\bar{k}_q}}^{k_p}
)
= 0$
Assume the grid is constant, so $\partial_t {e_\bar{i}}^i = 0$:
$
\partial_t \cdot (
{U^{i_1 ... i_p}}_{k_1 ... k_q}
)
\cdot {e^{\bar{i}_1}}_{i_1} \cdot ... \cdot {e^{\bar{i}_p}}_{i_p}
\cdot {e_{\bar{k}_1}}^{k_1} \cdot ... \cdot {e_{\bar{k}_q}}^{k_p}
+ \nabla_{\bar{j}} \cdot (
{F^{\bar{j} i_1 ... i_p}}_{k_1 ... k_q}
\cdot {e^{\bar{i}_1}}_{i_1} \cdot ... \cdot {e^{\bar{i}_p}}_{i_p}
\cdot {e_{\bar{k}_1}}^{k_1} \cdot ... \cdot {e_{\bar{k}_q}}^{k_p}
)
= 0$
Note that the connection of this $\nabla_{\bar{j}}$ is associated with the coordinate metric and therefore is symmetric.
You could equivalently use the connection $\nabla_j$ associated with the anholonomic components, which is antisymmetric, and then you would not need to worry about rescaling the components (because that information would already be stored in the connection).
That would look like this:
$(
\partial_t \cdot (
{U^{i_1 ... i_p}}_{k_1 ... k_q}
)
+ \nabla_j \cdot {F^{j i_1 ... i_p}}_{k_1 ... k_q}
)
\cdot {e^{\bar{i}_1}}_{i_1} \cdot ... \cdot {e^{\bar{i}_p}}_{i_p}
\cdot {e_{\bar{k}_1}}^{k_1} \cdot ... \cdot {e_{\bar{k}_q}}^{k_p}
= 0$
Is it safe to integrate the normalized-coordinate components across the volume, or should the components be converted to coordinate components inside the integral and then converted back to non-coordinates outside the integral?
$
\partial_t \cdot (
{U^{i_1 ... i_p}}_{k_1 ... k_q}
)
+ \nabla_j \cdot {F^{j i_1 ... i_p}}_{k_1 ... k_q}
= 0$
...?
For $\pmatrix{0 \\ 0}$ case:
For scalar conservation laws you don't have to worry about renormalizing indexes.
Also in this case the divergence turns out to be $\nabla F = \frac{1}{V} \partial (V F)$,
where $V = det({e_i}^a)$, the determinant of the derivative of the coordinate basis with the Cartesian basis (assuming the non-coordinate basis is normalized).
$\frac{\partial}{\partial t} U + \frac{1}{V} \partial_j (V F^j) = 0$
Next integrate the divergence over the volume.
$0 = \int_{t_1}^{t_2}
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\frac{\partial}{\partial t} U
+ \frac{1}{V} \partial_j (V F^j)
\right)
V
du_n ... du_1
dt
$
$0 = \int_{t_1}^{t_2}
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
V \frac{\partial}{\partial t} U
+ \partial_j (V F^j)
\right)
du_n ... du_1
dt
$
$0 =
\int_{t_1}^{t_2}
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
V \frac{\partial}{\partial t} U
\right)
du_n ... du_1
dt
+ \int_{t_1}^{t_2}
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\partial_j (V F^j)
du_n ... du_1
dt
$
Assume the grid volume and the flux doesn't change over time.
Assume U is constant throughout the region
$0 =
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
V
du_n ... du_1
\cdot
\left(
U|_{t=t_2} - U|_{t=t_1}
\right)
+ (t_2 - t_1)
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\partial_j (V F^j)
du_n ... du_1
$
Let $\mathcal{V} =
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
V
du_n ... du_1
$
Let $\Delta t = t_2 - t_1$
$U|_{t=t_2} =
U|_{t=t_1}
-
\frac{\Delta t}{\mathcal{V}}
\int_{u_{1,L}}^{u_{1,R}}
\overset{\hat{j}}{...}
\int_{u_{n,L}}^{u_{n,R}}
\left(
(V F^j)|_{u_j=u_{j,R}}
- (V F^j)|_{u_j=u_{j,L}}
\right)
du_n \overset{\hat{j}}{...} du_1
$
Assume the flux is constant across the surface:
$U|_{t=t_2} =
U|_{t=t_1}
-
\frac{\Delta t}{\mathcal{V}}
\int_{u_{1,L}}^{u_{1,R}}
\overset{\hat{j}}{...}
\int_{u_{n,L}}^{u_{n,R}}
\left(
(V F^j)|_{u_j=u_{j,R}} )
- (V F^j)|_{u_j=u_{j,L}} )
\right)
du_n \overset{\hat{j}}{...} du_1
$
Assume $F^j$ is constant throughout the surfaces $u = u_{j,L}$ and $u = u_{j,R}$:
$U|_{t=t_2} =
U|_{t=t_1}
-
\frac{\Delta t}{\mathcal{V}}
\left(
F^j|_{u_j=u_{j,R}}
\int_{u_{1,L}}^{u_{1,R}}
\overset{\hat{j}}{...}
\int_{u_{n,L}}^{u_{n,R}}
V|_{u_j=u_{j,R}}
du_n \overset{\hat{j}}{...} du_1
- F^j|_{u_j=u_{j,L}}
\int_{u_{1,L}}^{u_{1,R}}
\overset{\hat{j}}{...}
\int_{u_{n,L}}^{u_{n,R}}
V|_{u_j=u_{j,L}}
du_n \overset{\hat{j}}{...} du_1
\right)
$
Let $\Sigma_j = \int_{u_{1,L}}^{u_{1,R}}
\overset{\hat{j}}{...}
\int_{u_{n,L}}^{u_{n,R}}
V|_{u_j=u_{j,R}}
du_n \overset{\hat{j}}{...} du_1
$
$U|_{t=t_2} =
U|_{t=t_1}
-
\frac{\Delta t}{\mathcal{V}}
\left(
(F^j \Sigma_j)|_{u_j=u_{j,R}}
- (F^j \Sigma_j)|_{u_j=u_{j,L}}
\right)
$
Now for specific coordinate system examples of $\pmatrix{0 \\ 0}$:
Cartesian:
$g_{ij} = \delta_{ij}$
${e_{\bar{i}}}^i = \delta_\bar{i}^i$
$V = 1$
$\mathcal{V} = \Pi_{k=1}^n \Delta u_k$
$\Sigma_j = \Pi_{k=1,k\ne j}^n \Delta u_k$
$U|_{t=t_2} =
U|_{t=t_1}
-
\frac{\Delta t}{
\Pi_{k=1}^n (\Delta u_k)
}
\left(
(
F^j
\Pi_{k=1,k\ne j}^n (\Delta u_k)
)|_{u_j=u_{j,R}}
- (
F^j
\Pi_{k=1,k\ne j}^n (\Delta u_k)
)|_{u_j=u_{j,L}}
\right)
$
$U|_{t=t_2} = U|_{t=t_1} - \frac{\Delta t}{\Delta u_j} \left( F^j|_{u_j=u_{j,R}} - F^j|_{u_j=u_{j,L}} \right)$
Polar, $u = \{ r, \phi \}$
$g_{\bar{i}\bar{j}} = diag(1, r^2)$
${e_{\bar{i}}}^i = diag(1, r)$
$V = r$
$\mathcal{V} = \frac{1}{2} (r_R^2 - r_L^2) (\phi_R - \phi_L) = \frac{1}{2} \Delta (r^2) \Delta \phi$
$\Sigma_r = \int_{\phi=\phi_L}^{\phi=\phi_R} r d\phi = r (\phi_R - \phi_L) = r \Delta \phi$
$\Sigma_\phi = \int_{r=r_L}^{r=r_R} r dr = \frac{1}{2} (r_R^2 - r_L^2) = \frac{1}{2} \Delta (r^2)$
$U|_{t=t_2} =
U|_{t=t_1}
-
\frac{\Delta t}{
\frac{1}{2} \Delta (r^2) \Delta \phi
}
\left(
F^r|_{u_r=u_{r,R}} \cdot r_R \Delta \phi
- F^r|_{u_r=u_{r,L}} \cdot r_L \Delta \phi
+ F^\phi|_{u_\phi=u_{\phi,R}} \cdot \frac{1}{2} \Delta (r^2)
- F^\phi|_{u_\phi=u_{\phi,L}} \cdot \frac{1}{2} \Delta (r^2)
\right)
$
$U|_{t=t_2} =
U|_{t=t_1}
- \Delta t
\left(
\frac{2}{ \Delta (r^2) }
\left(
F^r|_{u_r=u_{r,R}} \cdot r_R
- F^r|_{u_r=u_{r,L}} \cdot r_L
\right)
+
\frac{1}{ \Delta \phi }
\left(
F^\phi|_{u_\phi=u_{\phi,R}}
- F^\phi|_{u_\phi=u_{\phi,L}}
\right)
\right)
$
Spherical, $u = \{r, \theta, \phi\}$
$g_{\bar{i}\bar{j}} = diag(1, r^2, r^2 sin(\theta)^2)$
${e_{\bar{i}}}^i = diag(1, r, r sin(\theta))$
$V = r^2 sin(\theta)$
$\mathcal{V} = \frac{1}{3} (r_R^3 - r_L^3) (\phi_R - \phi_L) (cos(\theta_R) - cos(\theta_L)) = \frac{1}{3} \Delta (r^3) \Delta cos(\theta)$
$\Sigma_r = \int_{\theta=\theta_L}^{\theta=\theta_R} \int_{\phi=\phi_L}^{\phi=\phi_R} r^2 sin(\theta) d\phi d\theta = r^2 (cos(\theta_R) - cos(\theta_L)) (\phi_R - \phi_L) = r^2 \Delta cos(\theta) \Delta \phi$
$\Sigma_\theta = \int_{r=r_L}^{r=r_R} \int_{\phi=\phi_L}^{\phi=\phi_R} r^2 sin(\theta) d\phi dr = \frac{1}{3} (r_R^3 - r_L^3) sin(\theta) (\phi_R - \phi_L) = \frac{1}{3} \Delta (r^3) sin(\theta) \Delta \phi$
$\Sigma_\phi = \int_{r=r_L}^{r=r_R} \int_{\theta=\theta_L}^{\theta=\theta_R} r^2 sin(\theta) d\theta dr = \frac{1}{3} (r_R^3 - r_L^3) (cos(\theta_R) - cos(\theta_L)) = \frac{1}{3} \Delta (r^3) \Delta cos(\theta)$
$U|_{t=t_2} =
U|_{t=t_1}
-
\frac{\Delta t}{\mathcal{V}}
\left(
(F^r \Sigma_r)|_{u_r=u_{r,R}}
- (F^r \Sigma_r)|_{u_r=u_{r,L}}
+ (F^\theta \Sigma_\theta)|_{u_\theta=u_{\theta,R}}
- (F^\theta \Sigma_\theta)|_{u_\theta=u_{\theta,L}}
+ (F^\phi \Sigma_\phi)|_{u_\phi=u_{\phi,R}}
- (F^\phi \Sigma_\phi)|_{u_\phi=u_{\phi,L}}
\right)
$
TODO fix this: $\frac{1}{V} \partial_j V$ is not correct for all $\pmatrix{p \\ q}$ tensors.
$0 = \int_{t_1}^{t_2}
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\frac{\partial}{\partial t}({U^{i_1 ... i_p}}_{k_1 ... k_q})
+ \frac{1}{V} (
\partial_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q} V)
)
\right)
V
du_n ... du_1
dt
$
assuming the grid is not dependent on time ($\frac{dV}{dt} = 0$)
$0 =
\int_{t_1}^{t_2}
\frac{\partial}{\partial t}
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
{U^{i_1 ... i_p}}_{k_1 ... k_q}
V
\right)
du_n ... du_1
\right)
dt
+
\int_{t_1}^{t_2}
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\partial_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q} V)
\right)
du_n ... du_1
dt
$
$0 =
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
{U^{i_1 ... i_p}}_{k_1 ... k_q}
V
\right)
du_n ... du_1
\right)|^{t_2}_{t_1}
+
\int_{t_1}^{t_2}
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\partial_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q} V)
\right)
du_n ... du_1
dt
$
assuming the flux is not dependent on time:
$0 =
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
{U^{i_1 ... i_p}}_{k_1 ... k_q}
V
\right)
du_n ... du_1
\right)|^{t_2}_{t_1}
+
\int_{t_1}^{t_2}
dt
\cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\partial_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q} V)
\right)
du_n ... du_1
$
$0 =
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
{U^{i_1 ... i_p}}_{k_1 ... k_q}
V
\right)
du_n ... du_1
\right)|^{t_2}_{t_1}
+
(t_2 - t_1) \cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\partial_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q} V)
\right)
du_n ... du_1
$
assuming the state is not dependent on space -- that it is constant throughout the integral region
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}|^{t_2}_{t_1}
\cdot
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
V
du_n ... du_1
\right)
=
-
(t_2 - t_1) \cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\partial_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q} V)
\right)
du_n ... du_1
$
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}|^{t_2}_{t_1}
=
- \frac{
(t_2 - t_1)
}{
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
V
du_n ... du_1
\right)
} \cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\partial_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q} V)
\right)
du_n ... du_1
$
(But why would you integrate the volume over the volume? Shouldn't you omit the $V$?)
For coordinate basis, ${e_i}^a = \delta_i^a$ and V = 1...
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
du_n ... du_1
\right)
} \cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\partial_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q})
\right)
du_n ... du_1
$
For a Cartesian basis, where $\nabla = \partial$ and $g_{ab} =\delta_{ab}$
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
du_n ... du_1
\right)
} \cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\partial_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q})
\right)
du_n ... du_1
$
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
du_n ... du_1
\right)
} \cdot
\Sigma_{j=1}^n
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{j-1,L}}^{u_{j-1,R}}
\int_{u_{j+1,L}}^{u_{j+1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\int_{u_{j,L}}^{u_{j,R}}
\left(
\partial_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q})
\right)
du_j du_n ... du_{j+1} du_{j-1} .. du_1
$
...using the FTC ...
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\Pi_{j=1}^n
\left(
u_{j,R} - u_{j,L}
\right)
} \cdot
\Sigma_{j=1}^n
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{j-1,L}}^{u_{j-1,R}}
\int_{u_{j+1,L}}^{u_{j+1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
{F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,R})
- {F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,L})
\right)
du_n ... du_{j+1} du_{j-1} .. du_1
$
Assuming the flux is not dependent on space:
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\Pi_{j=1}^n
\left(
u_{j,R} - u_{j,L}
\right)
} \cdot
\Sigma_{j=1}^n
\left(
{F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,R})
- {F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,L})
\right)
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{j-1,L}}^{u_{j-1,R}}
\int_{u_{j+1,L}}^{u_{j+1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
du_n ... du_{j+1} du_{j-1} .. du_1
$
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\Pi_{j=1}^n
\left(
u_{j,R} - u_{j,L}
\right)
} \cdot
\Sigma_{j=1}^n
\left(
\left(
{F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,R})
- {F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,L})
\right)
\cdot
\Pi_{k=1,k\ne j}^{k=n}
\left(
u_{k,R} - u_{k,L}
\right)
\right)
$
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- (t_2 - t_1)
\cdot
\Sigma_{j=1}^n
\left(
\frac{
1
}{
u_{j,R} - u_{j,L}
}
\cdot
\left(
{F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,R})
- {F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,L})
\right)
\right)
$
Go back before the assumption of $\nabla = \partial$
...
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
du_n ... du_1
\right)
} \cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\nabla_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q})
\right)
du_n ... du_1
$
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\Pi_{j=1}^n
\left(
u_{j,R} - u_{j,L}
\right)
} \cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\nabla_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q})
\right)
du_n ... du_1
$
...expand covariant derivative into connections...
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\Pi_{j=1}^n
\left(
u_{j,R} - u_{j,L}
\right)
} \cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\partial_j \left( {F^{j i_1 ... i_p}}_{k_1 ... k_q} \right)
+ \Sigma_l \left( {F^{j i_1 ... i_{l-1} m i_{l+1} ... i_p}}_{k_1 ... k_q} {\Gamma^{i_l}}_{mj} \right)
- \Sigma_l \left( {F^{j i_1 ... i_p}}_{k_1 ... k_{l-1} m k_{l+1} ... k_q} {\Gamma^m}_{k_l j} \right)
\right)
du_n ... du_1
$
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\Pi_{j=1}^n
\left(
u_{j,R} - u_{j,L}
\right)
} \cdot
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\partial_j \left( {F^{j i_1 ... i_p}}_{k_1 ... k_q} \right)
\right)
du_n ... du_1
+ \int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\Sigma_l \left( {F^{j i_1 ... i_{l-1} m i_{l+1} ... i_p}}_{k_1 ... k_q} {\Gamma^{i_l}}_{mj} \right)
\right)
du_n ... du_1
- \int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\Sigma_l \left( {F^{j i_1 ... i_p}}_{k_1 ... k_{l-1} m k_{l+1} ... k_q} {\Gamma^m}_{k_l j} \right)
\right)
du_n ... du_1
\right)
$
...using the FTC and assuming flux is not dependent on space...
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\Pi_{j=1}^n
\left(
u_{j,R} - u_{j,L}
\right)
} \cdot
\left(
\Sigma_{j=1}^n
\left(
\left(
{F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,R})
- {F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,L})
\right)
\cdot
\Pi_{k=1,k\ne j}^{k=n}
\left(
u_{k,R} - u_{k,L}
\right)
\right)
+ \int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\Sigma_l \left( {F^{j i_1 ... i_{l-1} m i_{l+1} ... i_p}}_{k_1 ... k_q} {\Gamma^{i_l}}_{mj} \right)
\right)
du_n ... du_1
- \int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\Sigma_l \left( {F^{j i_1 ... i_p}}_{k_1 ... k_{l-1} m k_{l+1} ... k_q} {\Gamma^m}_{k_l j} \right)
\right)
du_n ... du_1
\right)
$
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- (t_2 - t_1)
\Sigma_{j=1}^n
\left(
\frac{
1
}{
u_{j,R} - u_{j,L}
} \cdot
\left(
{F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,R})
- {F^{j i_1 ... i_p}}_{k_1 ... k_q}(u_j = u_{j,L})
\right)
+
\frac{
1
}{
\Pi_{j=1}^n
\left(
u_{j,R} - u_{j,L}
\right)
} \cdot
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\Sigma_l \left( {F^{j i_1 ... i_{l-1} m i_{l+1} ... i_p}}_{k_1 ... k_q} {\Gamma^{i_l}}_{mj} \right)
\right)
du_n ... du_1
- \int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\Sigma_l \left( {F^{j i_1 ... i_p}}_{k_1 ... k_{l-1} m k_{l+1} ... k_q} {\Gamma^m}_{k_l j} \right)
\right)
du_n ... du_1
\right)
\right)
$
Now go back to when we are using a non-coordinate system...
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}|^{t_2}_{t_1}
=
- (t_2 - t_1)
\frac{
1
}{
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
V
du_n ... du_1
\right)
} \cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
\nabla_j ({F^{j i_1 ... i_p}}_{k_1 ... k_q} V)
\right)
du_n ... du_1
$
...expand covariant derivative into connections...
$
{U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_2)
- {U^{i_1 ... i_p}}_{k_1 ... k_q}(t=t_1)
=
- \frac{
(t_2 - t_1)
}{
\left(
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
V
du_n ... du_1
\right)
} \cdot
\int_{u_{1,L}}^{u_{1,R}}
...
\int_{u_{n,L}}^{u_{n,R}}
\left(
{e^\bar{j}}_j \partial_\bar{j} \left( {F^{j i_1 ... i_p}}_{k_1 ... k_q} V \right)
+ \Sigma_l \left( {F^{j i_1 ... i_{l-1} m i_{l+1} ... i_p}}_{k_1 ... k_q} {\Gamma^{i_l}}_{mj} V \right)
- \Sigma_l \left( {F^{j i_1 ... i_p}}_{k_1 ... k_{l-1} m k_{l+1} ... k_q} {\Gamma^m}_{k_l j} V \right)
\right)
du_n ... du_1
$
Specific examples:
Cylindrical holonomic (grid metric)
connections of the holonomic basis:
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma}^{\phi}}_{\phi}}_r} = {\frac{1}{r}}$
; ${{{{ \Gamma}^{\phi}}_r}_{\phi}} = {\frac{1}{r}}$
; ${{{{ \Gamma}^r}_{\phi}}_{\phi}} = {-{r}}$
Cylindrical, anholonomic, normalized basis
${e_\bar{j}}^j = \left[ \begin{matrix}
1 & 0 & 0 \\
0 & r & 0 \\
0 & 0 & 1
\end{matrix} \right]$
${e^\bar{j}}_j = \left[ \begin{matrix}
1 & 0 & 0 \\
0 & \frac{1}{r} & 0 \\
0 & 0 & 1
\end{matrix} \right]$
$V = det({e_\bar{j}}^j) = r$
connections of the anholonomic normalized basis:
commutation coefficients:
${{{{ c}_{\hat{\theta}}}_r}^{\hat{\theta}}} = {\frac{1}{r}}$
; ${{{{ c}_r}_{\hat{\theta}}}^{\hat{\theta}}} = {-{\frac{1.0}{r}}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma}^{\hat{\theta}}}_{\hat{\theta}}}_r} = {\frac{1}{r}}$
; ${{{{ \Gamma}^r}_{\hat{\theta}}}_{\hat{\theta}}} = {-{\frac{1.0}{r}}}$
Spherical, anholonomic, normalized
${e_\bar{j}}^j = \left[ \begin{matrix}
1 & 0 & 0 \\
0 & r & 0 \\
0 & 0 & r sin (\theta)
\end{matrix} \right]$
${e^\bar{j}}_j = \left[ \begin{matrix}
1 & 0 & 0 \\
0 & \frac{1}{r} & 0 \\
0 & 0 & \frac{1}{r sin(\theta)}
\end{matrix} \right]$
$V = det({e_\bar{j}}^j) = r^2 sin(\theta)$
connections of the anholonomic normalized basis:
${\Gamma^r}_{\phi \phi} = -\frac{1}{r}$
${\Gamma^r}_{\theta \theta} = -\frac{1}{r}$
${\Gamma^\theta}_{\theta r} = \frac{1}{r}$
${\Gamma^\theta}_{\phi \phi} = -\frac{1}{r} \frac{cos(\theta)}{sin(\theta)}$
${\Gamma^\phi}_{\phi r} = \frac{1}{r}$
${\Gamma^\phi}_{\phi \theta} = \frac{1}{r} \frac{cos(\theta)}{sin(\theta)}$
commutation coefficients:
${{{{ c}_{\hat{\theta}}}_r}^{\hat{\theta}}} = {\frac{1}{r}}$
; ${{{{ c}_r}_{\hat{\theta}}}^{\hat{\theta}}} = {-{\frac{1}{r}}}$
; ${{{{ c}_{\hat{\phi}}}_r}^{\hat{\phi}}} = {\frac{1}{r}}$
; ${{{{ c}_{\hat{\phi}}}_{\hat{\theta}}}^{\hat{\phi}}} = {\frac{cos\left( \theta\right)}{{{r}} {{sin\left( \theta\right)}}}}$
; ${{{{ c}_r}_{\hat{\phi}}}^{\hat{\phi}}} = {-{\frac{1.0}{r}}}$
; ${{{{ c}_{\hat{\theta}}}_{\hat{\phi}}}^{\hat{\phi}}} = {\frac{-{cos\left( \theta\right)}}{{{r}} {{sin\left( \theta\right)}}}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma}^{\hat{\theta}}}_{\hat{\theta}}}_r} = {\frac{1}{r}}$
; ${{{{ \Gamma}^{\hat{\phi}}}_{\hat{\phi}}}_r} = {\frac{1}{r}}$
; ${{{{ \Gamma}^r}_{\hat{\theta}}}_{\hat{\theta}}} = {-{\frac{1.0}{r}}}$
; ${{{{ \Gamma}^{\hat{\phi}}}_{\hat{\phi}}}_{\hat{\theta}}} = {\frac{cos\left( \theta\right)}{{{r}} {{sin\left( \theta\right)}}}}$
; ${{{{ \Gamma}^r}_{\hat{\phi}}}_{\hat{\phi}}} = {-{\frac{1.0}{r}}}$
; ${{{{ \Gamma}^{\hat{\theta}}}_{\hat{\phi}}}_{\hat{\phi}}} = {\frac{-{cos\left( \theta\right)}}{{{r}} {{sin\left( \theta\right)}}}}$
for the scalar case...
$0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
\frac{\partial}{\partial t}(U)
+
\frac{1}{
r^2 sin(\theta)
} (
\nabla_j (
r^2 sin(\theta)
F^j
)
)
\right)
r^2 sin(\theta)
d\phi
d\theta
dr
dt
$
$0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta)
\frac{\partial}{\partial t}(U)
+ \nabla_r (
r^2 sin(\theta)
F^r
)
+ \nabla_\theta (
r^2 sin(\theta)
F^\theta
)
+ \nabla_\phi (
r^2 sin(\theta)
F^\phi
)
\right)
d\phi
d\theta
dr
dt
$
$0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta)
\frac{\partial}{\partial t}(U)
+ sin(\theta) \partial_r (
r^2
F^r
)
+ r^2 \partial_\theta (
sin(\theta)
F^\theta
)
+ r^2 sin(\theta) \partial_\phi (
F^\phi
)
\right)
d\phi
d\theta
dr
dt
$
$0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta)
\frac{\partial}{\partial t}(U)
\right)
d\phi
d\theta
dr
dt
+
\int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
sin(\theta) \partial_r (
r^2
F^r
)
\right)
d\phi
d\theta
dr
dt
+
\int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 \partial_\theta (
sin(\theta)
F^\theta
)
\right)
d\phi
d\theta
dr
dt
+
\int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \partial_\phi (
F^\phi
)
\right)
d\phi
d\theta
dr
dt
$
$0 =
\int_{\phi_L}^{\phi_R} d\phi
\cdot
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{t_1}^{t_2}
\frac{\partial}{\partial t}(U)
dt
r^2 sin(\theta)
d\theta
dr
+
\int_{t_1}^{t_2} dt
\cdot
\int_{\phi_L}^{\phi_R} d\phi
\cdot
\int_{\theta_L}^{\theta_R}
\int_{r_L}^{r_R}
\partial_r (
r^2
F^r
)
dr
sin(\theta)
d\theta
+
\int_{t_1}^{t_2} dt
\cdot
\int_{\phi_L}^{\phi_R} d\phi
\cdot
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\partial_\theta (
sin(\theta)
F^\theta
)
d\theta
r^2
dr
+
\int_{t_1}^{t_2} dt
\cdot
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\partial_\phi (
F^\phi
)
d\phi
sin(\theta)
d\theta
r^2
dr
$
$0 =
(U|_{t=t_2} - U|_{t=t_1})
\cdot \frac{1}{3} ( (r_R)^3 - (r_L)^3 )
\cdot (-cos(\theta_R) + cos(\theta_L))
\cdot (\phi_R - \phi_L)
+
(t_2 - t_1)
\cdot ( (r_R)^2 F^r|_{r=r_R} - (r_L)^2 F^r|_{r=r_L} )
\cdot (-cos(\theta_R) + cos(\theta_L))
\cdot (\phi_R - \phi_L)
+
(t_2 - t_1)
\cdot \frac{1}{3} ( (r_R)^3 - (r_L)^3 )
\cdot (
sin(\theta_R) F^\theta|_{\theta=\theta_R}
- sin(\theta_L) F^\theta|_{\theta=\theta_L}
)
\cdot (\phi_R - \phi_L)
+
(t_2 - t_1)
\cdot \frac{1}{3} ( (r_R)^3 - (r_L)^3 )
\cdot (-cos(\theta_R) + cos(\theta_L))
\cdot (
F^\phi|_{\phi=\phi_R}
- F^\phi|_{\phi=\phi_L}
)
$
$U|_{t=t_2} =
U|_{t=t_1}
-
(t_2 - t_1)
\left(
\frac{
(r_R)^2 F^r|_{r=r_R} - (r_L)^2 F^r|_{r=r_L}
}{
\frac{1}{3} ( (r_R)^3 - (r_L)^3 )
}
+
\frac{
sin(\theta_R) F^\theta|_{\theta=\theta_R}
- sin(\theta_L) F^\theta|_{\theta=\theta_L}
}{
cos(\theta_L) - cos(\theta_R)
}
+
\frac{
F^\phi|_{\phi=\phi_R}
- F^\phi|_{\phi=\phi_L}
}{
\phi_R - \phi_L
}
\right)
$
now for the vector case...
$0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta)
\frac{\partial}{\partial t}(U^i)
+ \nabla_r (
r^2 sin(\theta)
F^{r i}
)
+ \nabla_\theta (
r^2 sin(\theta)
F^{\theta i}
)
+ \nabla_\phi (
r^2 sin(\theta)
F^{\phi i}
)
\right)
d\phi
d\theta
dr
dt
$
$0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^i)
+ \partial_r ( r^2 sin(\theta) F^{r i} )
+ r^2 sin(\theta) F^{r k} {\Gamma^i}_{r k}
+ \partial_\theta ( r^2 sin(\theta) F^{\theta i} )
+ r^2 sin(\theta) F^{\theta k} {\Gamma^i}_{\theta k}
+ \partial_\phi ( r^2 sin(\theta) F^{\phi i} )
+ r^2 sin(\theta) F^{\phi k} {\Gamma^i}_{\phi k}
\right)
d\phi
d\theta
dr
dt
$
$0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^i)
+ \partial_r ( r^2 sin(\theta) F^{r i} )
+ r^2 sin(\theta) (F^{r r} {\Gamma^i}_{r r} + F^{r \theta} {\Gamma^i}_{r \theta} + F^{r \phi} {\Gamma^i}_{r \phi} )
+ \partial_\theta ( r^2 sin(\theta) F^{\theta i} )
+ r^2 sin(\theta) (F^{\theta r} {\Gamma^i}_{\theta r} + F^{\theta \theta} {\Gamma^i}_{\theta \theta} + F^{\theta \phi} {\Gamma^i}_{\theta \phi})
+ \partial_\phi ( r^2 sin(\theta) F^{\phi i} )
+ r^2 sin(\theta) (F^{\phi r} {\Gamma^i}_{\phi r} + F^{\phi \theta} {\Gamma^i}_{\phi \theta} + F^{\phi \phi} {\Gamma^i}_{\phi \phi})
\right)
d\phi
d\theta
dr
dt
$
$\left\{ \begin{matrix}
0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^r)
+ \partial_r ( r^2 sin(\theta) F^{r r} )
+ r^2 sin(\theta) (F^{r r} {\Gamma^r}_{r r} + F^{r \theta} {\Gamma^r}_{r \theta} + F^{r \phi} {\Gamma^r}_{r \phi} )
+ \partial_\theta ( r^2 sin(\theta) F^{\theta r} )
+ r^2 sin(\theta) (F^{\theta r} {\Gamma^r}_{\theta r} + F^{\theta \theta} {\Gamma^r}_{\theta \theta} + F^{\theta \phi} {\Gamma^r}_{\theta \phi})
+ \partial_\phi ( r^2 sin(\theta) F^{\phi r} )
+ r^2 sin(\theta) (F^{\phi r} {\Gamma^r}_{\phi r} + F^{\phi \theta} {\Gamma^r}_{\phi \theta} + F^{\phi \phi} {\Gamma^r}_{\phi \phi})
\right)
d\phi
d\theta
dr
dt
\\
0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^\theta)
+ \partial_r ( r^2 sin(\theta) F^{r \theta} )
+ r^2 sin(\theta) (F^{r r} {\Gamma^\theta}_{r r} + F^{r \theta} {\Gamma^\theta}_{r \theta} + F^{r \phi} {\Gamma^\theta}_{r \phi} )
+ \partial_\theta ( r^2 sin(\theta) F^{\theta \theta} )
+ r^2 sin(\theta) (F^{\theta r} {\Gamma^\theta}_{\theta r} + F^{\theta \theta} {\Gamma^\theta}_{\theta \theta} + F^{\theta \phi} {\Gamma^\theta}_{\theta \phi})
+ \partial_\phi ( r^2 sin(\theta) F^{\phi \theta} )
+ r^2 sin(\theta) (F^{\phi r} {\Gamma^\theta}_{\phi r} + F^{\phi \theta} {\Gamma^\theta}_{\phi \theta} + F^{\phi \phi} {\Gamma^\theta}_{\phi \phi})
\right)
d\phi
d\theta
dr
dt
\\
0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^\phi)
+ \partial_r ( r^2 sin(\theta) F^{r \phi} )
+ r^2 sin(\theta) (F^{r r} {\Gamma^\phi}_{r r} + F^{r \theta} {\Gamma^\phi}_{r \theta} + F^{r \phi} {\Gamma^\phi}_{r \phi} )
+ \partial_\theta ( r^2 sin(\theta) F^{\theta \phi} )
+ r^2 sin(\theta) (F^{\theta r} {\Gamma^\phi}_{\theta r} + F^{\theta \theta} {\Gamma^\phi}_{\theta \theta} + F^{\theta \phi} {\Gamma^\phi}_{\theta \phi})
+ \partial_\phi ( r^2 sin(\theta) F^{\phi \phi} )
+ r^2 sin(\theta) (F^{\phi r} {\Gamma^\phi}_{\phi r} + F^{\phi \theta} {\Gamma^\phi}_{\phi \theta} + F^{\phi \phi} {\Gamma^\phi}_{\phi \phi})
\right)
d\phi
d\theta
dr
dt
\\
\end{matrix} \right\}$
$\left\{ \begin{matrix}
0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^r)
+ \partial_r ( r^2 sin(\theta) F^{r r} )
+ \partial_\theta ( r^2 sin(\theta) F^{\theta r} )
+ \partial_\phi ( r^2 sin(\theta) F^{\phi r} )
+ r^2 sin(\theta) (-F^{\theta \theta} \frac{1}{r})
+ r^2 sin(\theta) (-F^{\phi \phi} \frac{1}{r})
\right)
d\phi
d\theta
dr
dt
\\
0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^\theta)
+ \partial_r ( r^2 sin(\theta) F^{r \theta} )
+ \partial_\theta ( r^2 sin(\theta) F^{\theta \theta} )
+ \partial_\phi ( r^2 sin(\theta) F^{\phi \theta} )
+ r^2 sin(\theta) (F^{\theta r} \frac{1}{r})
+ r^2 sin(\theta) (-F^{\phi \phi} \frac{cos(\theta)}{r sin(\theta)})
\right)
d\phi
d\theta
dr
dt
\\
0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^\phi)
+ \partial_r ( r^2 sin(\theta) F^{r \phi} )
+ \partial_\theta ( r^2 sin(\theta) F^{\theta \phi} )
+ \partial_\phi ( r^2 sin(\theta) F^{\phi \phi} )
+ r^2 sin(\theta) (F^{\phi r} \frac{1}{r} + F^{\phi \theta} \frac{cos(\theta)}{r sin(\theta)})
\right)
d\phi
d\theta
dr
dt
\\
\end{matrix} \right\}$
$\left\{ \begin{matrix}
0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^r)
+ \partial_r ( r^2 sin(\theta) F^{r r} )
+ \partial_\theta ( r^2 sin(\theta) F^{\theta r} )
+ \partial_\phi ( r^2 sin(\theta) F^{\phi r} )
- r sin(\theta) F^{\theta \theta}
- r sin(\theta) F^{\phi \phi}
\right)
d\phi
d\theta
dr
dt
\\
0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^\theta)
+ \partial_r ( r^2 sin(\theta) F^{r \theta} )
+ \partial_\theta ( r^2 sin(\theta) F^{\theta \theta} )
+ \partial_\phi ( r^2 sin(\theta) F^{\phi \theta} )
+ r sin(\theta) F^{\theta r}
- r cos(\theta) F^{\phi \phi}
\right)
d\phi
d\theta
dr
dt
\\
0 = \int_{t_1}^{t_2}
\int_{r_L}^{r_R}
\int_{\theta_L}^{\theta_R}
\int_{\phi_L}^{\phi_R}
\left(
r^2 sin(\theta) \frac{\partial}{\partial t}(U^\phi)
+ \partial_r ( r^2 sin(\theta) F^{r \phi} )
+ \partial_\theta ( r^2 sin(\theta) F^{\theta \phi} )
+ \partial_\phi ( r^2 sin(\theta) F^{\phi \phi} )
+ r sin(\theta) F^{\phi r}
+ r cos(\theta) F^{\phi \theta}
\right)
d\phi
d\theta
dr
dt
\\
\end{matrix} \right\}$
$\left\{ \begin{matrix}
0 =
(U^r|_{t=t_2} - U^r|_{t=t_1}) \cdot (\frac{1}{3} ( (r_R)^3 - (r_L)^3 )) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (\phi_R - \phi_L)
+ (t_2 - t_1) \cdot ( (r_R)^2 F^{rr}|_{r=r_R} - (r_L)^2 F^{rr}|_{r=r_L} ) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (\phi_R - \phi_L)
+ (t_2 - t_1) \cdot (\frac{1}{3} ( (r_R)^3 - (r_L)^3 )) \cdot (sin(\theta_R) F^{\theta r}|_{\theta=\theta_R} - sin(\theta_L) F^{\theta r}|_{\theta=\theta_L}) \cdot (\phi_R - \phi_L)
+ (t_2 - t_1) \cdot (\frac{1}{3} ( (r_R)^3 - (r_L)^3 )) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (F^{\phi r}|_{\phi=\phi_R} - F^{\phi r}|_{\phi=\phi_L})
- (t_2 - t_1) \cdot (\frac{1}{2} ( (r_R)^2 - (r_L)^2 )) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (\phi_R - \phi_L) \cdot F^{\theta \theta}
- (t_2 - t_1) \cdot (\frac{1}{2} ( (r_R)^2 - (r_L)^2 )) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (\phi_R - \phi_L) \cdot F^{\phi \phi}
\\
0 =
(U^\theta|_{t=t_2} - U^\theta|_{t=t_1}) \cdot (\frac{1}{3} ( (r_R)^3 - (r_L)^3 )) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (\phi_R - \phi_L)
+ (t_2 - t_1) \cdot ( (r_R)^2 F^{r \theta}|_{r=r_R} - (r_L)^2 F^{r \theta}|_{r=r_L} ) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (\phi_R - \phi_L)
+ (t_2 - t_1) \cdot (\frac{1}{3} ( (r_R)^3 - (r_L)^3 )) \cdot (sin(\theta_R) F^{\theta \theta}|_{\theta=\theta_R} - sin(\theta_L) F^{\theta \theta}|_{\theta=\theta_L}) \cdot (\phi_R - \phi_L)
+ (t_2 - t_1) \cdot (\frac{1}{3} ( (r_R)^3 - (r_L)^3 )) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (F^{\phi \theta}|_{\phi=\phi_R} - F^{\phi \theta}|_{\phi=\phi_L})
+ (t_2 - t_1) \cdot (\frac{1}{2} ( (r_R)^2 - (r_L)^2 )) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (\phi_R - \phi_L) \cdot F^{\theta r}
- (t_2 - t_1) \cdot (\frac{1}{2} ( (r_R)^2 - (r_L)^2 )) \cdot (sin(\theta_R) - sin(\theta_L)) \cdot (\phi_R - \phi_L) \cdot F^{\phi \phi}
\\
0 =
(U^\phi|_{t=t_2} - U^\phi|_{t=t_1}) \cdot (\frac{1}{3} ( (r_R)^3 - (r_L)^3 )) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (\phi_R - \phi_L)
+ (t_2 - t_1) \cdot ( (r_R)^2 F^{r \phi}|_{r=r_R} - (r_L)^2 F^{r \phi}|_{r=r_L} ) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (\phi_R - \phi_L)
+ (t_2 - t_1) \cdot (\frac{1}{3} ( (r_R)^3 - (r_L)^3 )) \cdot (sin(\theta_R) F^{\theta \phi}|_{\theta=\theta_R} - sin(\theta_L) F^{\theta \phi}|_{\theta=\theta_L}) \cdot (\phi_R - \phi_L)
+ (t_2 - t_1) \cdot (\frac{1}{3} ( (r_R)^3 - (r_L)^3 )) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (F^{\phi \phi}|_{\phi=\phi_R} - F^{\phi \phi}|_{\phi=\phi_L})
+ (t_2 - t_1) \cdot (\frac{1}{2} ( (r_R)^2 - (r_L)^2 )) \cdot (-cos(\theta_R) + cos(\theta_L)) \cdot (\phi_R - \phi_L) \cdot F^{\phi r}
+ (t_2 - t_1) \cdot (\frac{1}{2} ( (r_R)^2 - (r_L)^2 )) \cdot (sin(\theta_R) - sin(\theta_L)) \cdot (\phi_R - \phi_L) \cdot F^{\phi \theta}
\end{matrix} \right\}$
$\left\{ \begin{matrix}
U^r|_{t=t_2}
= U^r|_{t=t_1}
- (t_2 - t_1) \cdot
\left(
\frac{ (r_R)^2 F^{rr}|_{r=r_R} - (r_L)^2 F^{rr}|_{r=r_L} }{ \frac{1}{3} ( (r_R)^3 - (r_L)^3 ) }
+ \frac{ sin(\theta_R) F^{\theta r}|_{\theta=\theta_R} - sin(\theta_L) F^{\theta r}|_{\theta=\theta_L} }{ -cos(\theta_R) + cos(\theta_L) }
+ \frac{ F^{\phi r}|_{\phi=\phi_R} - F^{\phi r}|_{\phi=\phi_L} }{ \phi_R - \phi_L }
- F^{\theta \theta}
- F^{\phi \phi}
\right)
\\
U^\theta|_{t=t_2}
= U^\theta|_{t=t_1}
- (t_2 - t_1) \cdot
\left(
\frac{ (r_R)^2 F^{r \theta}|_{r=r_R} - (r_L)^2 F^{r \theta}|_{r=r_L} }{ \frac{1}{3} ( (r_R)^3 - (r_L)^3 ) }
+ \frac{ sin(\theta_R) F^{\theta \theta}|_{\theta=\theta_R} - sin(\theta_L) F^{\theta \theta}|_{\theta=\theta_L} }{ -cos(\theta_R) + cos(\theta_L) }
+ \frac{ F^{\phi \theta}|_{\phi=\phi_R} - F^{\phi \theta}|_{\phi=\phi_L} }{ \phi_R - \phi_L }
+ F^{\theta r}
- \frac{ sin(\theta_R) - sin(\theta_L) }{ -cos(\theta_R) + cos(\theta_L) } \cdot F^{\phi \phi}
\right)
\\
U^\phi|_{t=t_2}
= U^\phi|_{t=t_1}
- (t_2 - t_1) \cdot
\left(
\frac{ (r_R)^2 F^{r \phi}|_{r=r_R} - (r_L)^2 F^{r \phi}|_{r=r_L} }{ \frac{1}{3} ( (r_R)^3 - (r_L)^3 ) }
+ \frac{ (sin(\theta_R) F^{\theta \phi}|_{\theta=\theta_R} - sin(\theta_L) F^{\theta \phi}|_{\theta=\theta_L}) }{ (-cos(\theta_R) + cos(\theta_L)) }
+ \frac{ (F^{\phi \phi}|_{\phi=\phi_R} - F^{\phi \phi}|_{\phi=\phi_L}) }{ \phi_R - \phi_L }
+ F^{\phi r}
+ \frac{ (sin(\theta_R) - sin(\theta_L)) }{ (-cos(\theta_R) + cos(\theta_L)) } \cdot F^{\phi \theta}
\right)
\end{matrix} \right\}$
$\left[ \begin{matrix}
U^r \\
U^\theta \\
U^\phi
\end{matrix} \right]_{t=t_2}
=
\left[ \begin{matrix}
U^r \\
U^\theta \\
U^\phi
\end{matrix} \right]_{t=t_1}
- (t_2 - t_1) \cdot
\left(
\frac{1}{
\frac{1}{3} ( (r_R)^3 - (r_L)^3 )
} \cdot
\left[ \begin{matrix}
(r_R)^2 F^{rr}|_{r=r_R} - (r_L)^2 F^{rr}|_{r=r_L}
\\
(r_R)^2 F^{r \theta}|_{r=r_R} - (r_L)^2 F^{r \theta}|_{r=r_L}
\\
(r_R)^2 F^{r \phi}|_{r=r_R} - (r_L)^2 F^{r \phi}|_{r=r_L}
\end{matrix} \right]
+
\frac{1}{
-cos(\theta_R) + cos(\theta_L)
} \cdot
\left[ \begin{matrix}
sin(\theta_R) F^{\theta r}|_{\theta=\theta_R} - sin(\theta_L) F^{\theta r}|_{\theta=\theta_L}
\\
sin(\theta_R) F^{\theta \theta}|_{\theta=\theta_R} - sin(\theta_L) F^{\theta \theta}|_{\theta=\theta_L}
\\
sin(\theta_R) F^{\theta \phi}|_{\theta=\theta_R} - sin(\theta_L) F^{\theta \phi}|_{\theta=\theta_L}
\end{matrix} \right]
+
\frac{1}{
\phi_R - \phi_L
} \cdot
\left[ \begin{matrix}
F^{\phi r}|_{\phi=\phi_R} - F^{\phi r}|_{\phi=\phi_L}
\\
F^{\phi \theta}|_{\phi=\phi_R} - F^{\phi \theta}|_{\phi=\phi_L}
\\
F^{\phi \phi}|_{\phi=\phi_R} - F^{\phi \phi}|_{\phi=\phi_L}
\end{matrix} \right]
+
\left[ \begin{matrix}
- F^{\theta \theta}
- F^{\phi \phi}
\\
F^{\theta r}
- \frac{ sin(\theta_R) - sin(\theta_L) }{ -cos(\theta_R) + cos(\theta_L) } \cdot F^{\phi \phi}
\\
F^{\phi r}
+ \frac{ (sin(\theta_R) - sin(\theta_L)) }{ (-cos(\theta_R) + cos(\theta_L)) } \cdot F^{\phi \theta}
\end{matrix} \right]
\right)
$
TODO check this out too:
https://www.groundai.com/project/fifth-order-finite-volume-weno-in-general-orthogonally-curvilinear-coordinates4982/1