3-NIRVANA-user-guide/3.2-User-interfaces.md

@@ -947,18 +947,6 @@ field is per se cell-wise divergence-free. For a grid cell
 (`ix`,`iy`,`iz`) on superblock `g` this means
 
 ![ic_b](uploads/411b3e817fec94afa0a2861772697e71/ic_b.png)  
-$$\\mathtt{g->bx\[iz\]\[iy\]\[ix\]}=\\frac{1}{\\delta\\,\\!\\mathcal{A}\_x}\\int
-\\limits\_\\mathtt{g->y\[iy\]}^\\mathtt{g->y\[iy+1\]}
-\\int\\limits\_\\mathtt{g->z\[iz\]}^\\mathtt{g->z\[iz+1\]} B\_x(\\mathtt{g->x\[ix\]},y,z)h\_yh\_zdydz$$
-
-$$\\mathtt{g->by\[iz\]\[iy\]\[ix\]}=\\frac{1}{\\delta\\,\\!\\mathcal{A}\_y}\\int
-\\limits\_\\mathtt{g->x\[ix\]}^\\mathtt{g->x\[ix+1\]}
-\\int\\limits\_\\mathtt{g->z\[iz\]}^\\mathtt{g->z\[iz+1\]} B\_y(x,\\mathtt{g->y\[iy\]},z)h\_zdxdz$$
-
-$$\\mathtt{g->bz\[iz\]\[iy\]\[ix\]}=\\frac{1}{\\delta\\,\\!\\mathcal{A}\_z}\\int
-\\limits\_\\mathtt{g->x\[ix\]}^\\mathtt{g->x\[ix+1\]}
-\\int\\limits\_\\mathtt{g->y\[iy\]}^\\mathtt{g->y\[iy+1\]} B\_z(x,y,\\mathtt{g->z\[iz\]})h\_ydxdy$$
-
 The numerical expressions for the cell face contents are:
 
 | cell face            | expression                               |
@@ -971,28 +959,19 @@ The numerical expressions for the cell face contents are:
 **B** = ∇ × **A**. When discretized in integral form this gives for the
 face-averaged magnetic field components
 
-`g->bx[iz][iy][ix]` = \[*h*<sub>*y*</sub>*Δ*<sub>`i`*y*</sub>(*h*<sub>*z**y*</sub>*Â*<sub>*z*</sub>)−*h*<sub>*y*</sub>*Δ*<sub>`i`*z*</sub>*Â*<sub>*y*</sub>\]/*δ*𝒜<sub>*x*</sub>
+`g->bx[iz][iy][ix]` = \[*h*<sub>*y*</sub>*Δ*<sub>`iy`</sub>(*h*<sub>*zy*</sub>*Â*<sub>*z*</sub>)−*h*<sub>*y*</sub>*Δ*<sub>`iz`</sub>*Â*<sub>*y*</sub>\]/*δ*𝒜<sub>*x*</sub>
 
-`g->by[iz][iy][ix]` = \[*Δ*<sub>`i`*z*</sub>*Â*<sub>*x*</sub>−*h*<sub>*z**y*</sub>*Δ*<sub>`i`*x*</sub>(*h*<sub>*y*</sub>*Â*<sub>*z*</sub>)\]/*δ*𝒜<sub>*y*</sub>
+`g->by[iz][iy][ix]` = \[*Δ*<sub>`iz`</sub>*Â*<sub>*x*</sub>−*h*<sub>*zy*</sub>*Δ*<sub>`ix`</sub>(*h*<sub>*y*</sub>*Â*<sub>*z*</sub>)\]/*δ*𝒜<sub>*y*</sub>
 
-`g->bz[iz][iy][ix]` = \[*Δ*<sub>`i`*x*</sub>(*h*<sub>*y*</sub>*Â*<sub>*y*</sub>)−*Δ*<sub>`i`*y*</sub>*Â*<sub>*x*</sub>\]/*δ*𝒜<sub>*z*</sub>
+`g->bz[iz][iy][ix]` = \[*Δ*<sub>`ix`</sub>(*h*<sub>*y*</sub>*Â*<sub>*y*</sub>)−*Δ*<sub>`iy`</sub>*Â*<sub>*x*</sub>\]/*δ*𝒜<sub>*z*</sub>
 
 where (*Â*<sub>*x*</sub>, *Â*<sub>*y*</sub>, *Â*<sub>*z*</sub>) denote
 the path integrals
-
-$$\\hat{A}\_x(\\mathtt{ix,iy,iz})=\\int\\limits\_\\mathtt{g-&gt;x\[ix\]}^\\mathtt{g-&gt;x\[ix+1\]}
-A\_x(x,\\mathtt{g-&gt;y\[iy\]},\\mathtt{g-&gt;z\[iz\]})dx$$
-
-$$\\hat{A}\_y(\\mathtt{ix,iy,iz})=\\int\\limits\_\\mathtt{g-&gt;y\[iy\]}^\\mathtt{g-&gt;y\[iy+1\]}
-A\_y(\\mathtt{g-&gt;x\[ix\]},y,\\mathtt{g-&gt;z\[iz\]})dy$$
-
-$$\\hat{A}\_z(\\mathtt{ix,iy,iz})=\\int\\limits\_\\mathtt{g-&gt;z\[iz\]}^\\mathtt{g-&gt;z\[iz+1\]}
-A\_z(\\mathtt{g-&gt;x\[ix\]},\\mathtt{g-&gt;y\[iy\]},z)dz$$
-
+![ic_A](uploads/d7670a3203d0f3a1e2a9a759e69e2e77/ic_A.png)  
 over corresponding cell edges. The difference operator
 *Δ*<sub>`i`*y*</sub>, for instance, is given by
 
-*Δ*<sub>`i`*y*</sub>*Â*<sub>*x*</sub> = *Â*<sub>*x*</sub>(`i``x``,` `i``y` `+` `1``,` `i``z`) − *Â*<sub>*x*</sub>(`i``x``,` `i``y``,` `i``z`)
+*Δ*<sub>`iy`</sub>*Â*<sub>*x*</sub> = *Â*<sub>*x*</sub>(`ix`,`iy`+1,` `iz`) − *Â*<sub>*x*</sub>(`ix`,`iy`,`iz`)
 
 and similarly for others. If the *Â*’s are *unambiguously* computed (or
 approximated) on coinciding cell edges in the grid hierarchy, the