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

@@ -950,16 +950,6 @@ read
 
 ![ic_b](uploads/411b3e817fec94afa0a2861772697e71/ic_b.png)  
 
-`g->bx[iz][iy][ix]`$=\frac{1}{\delta \mathcal{A}_x}\int
-\limits_\mathtt{g->y[iy]}^\mathtt{g->y[iy+1]}
-\int\limits_\mathtt{g->z[iz]}^\mathtt{g->z[iz+1]} B_x(\mathtt{g->x[ix]},y,z)h_yh_zdydz$
-
-$$\mathtt{g->by[iz][iy][ix]}=\frac{1}{\delta \mathcal{A}_y}\int
-\limits_\mathtt{g->x[ix]}^\mathtt{g->x[ix+1]}
-\int\limits_\mathtt{g->z[iz]}^\mathtt{g->z[iz+1]} B_y(x,\mathtt{g->y[iy]},z)h_zdxdz$$
-$$\mathtt{g->bz[iz][iy][ix]}=\frac{1}{\delta \mathcal{A}_z}\int
-\limits_\mathtt{g->x[ix]}^\mathtt{g->x[ix+1]}
-\int\limits_\mathtt{g->y[iy]}^\mathtt{g->y[iy+1]} B_z(x,y,\mathtt{g->z[iz]})h_ydxdy$$
 where the numerical expressions for the cell faces are:
 
 | cell face          | expression                               |
@@ -975,6 +965,16 @@ integration over cell faces turns out to be too complicated.
 The face-averaged magnetic field components are then obtained from the
 integral form of $\mathbf{B}=\nabla\times \mathbf{A}$:
 
+`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]` = \[*Δ*<sub>`iz`</sub>*Â*<sub>*x*</sub>−*h*<sub>*zy*</sub>*Δ*<sub>`ix`</sub>(*h*<sub>*y*</sub>*Â*<sub>*z*</sub>)\]/*δ*𝒜<sub>*y*</sub>
+
+`g->bz[iz][iy][ix]` = \[*Δ*<sub>`ix`</sub>(*h*<sub>*y*</sub>*Â*<sub>*y*</sub>)−*Δ*<sub>`iy`</sub>*Â*<sub>*x*</sub>\]/*δ*𝒜<sub>*z*</sub>
+
+where (*Â*<sub>*x*</sub>, *Â*<sub>*y*</sub>, *Â*<sub>*z*</sub>) denote
+the path integrals
+
+
 `g->bx[iz][iy][ix]`$=\left[h_y\Delta_{iy} (h_{zy}\hat{A}_z)
 -h_y\Delta_{iz} \hat{A}_y\right]/\delta \mathcal{A}_x$
 
@@ -995,6 +995,8 @@ $\Delta_{iz}X=X(\mathtt{iz,iy,ix+1})-X(\mathtt{iz,iy,ix})$
 
 and $(\hat{A}_x,\hat{A}_y,\hat{A}_z)$ are the cell-edge integrals
 
+![ic_A](uploads/d7670a3203d0f3a1e2a9a759e69e2e77/ic_A.png)  
+
 $$\hat{A}_x(\mathtt{iz,iy,ix})=\int\limits_\mathtt{g->x[ix]}^\mathtt{g->x[ix+1]}
 A_x(x,\mathtt{g->y[iy]},\mathtt{g->z[iz]})dx$$