Physical-quantities.md

@@ -34,7 +34,7 @@ E_{\textrm{pot.ext}} = \int \sum_{\sigma=\{\uparrow,\downarrow\}} V_{\sigma}^{\t
 ```
 * `E_pairext`:
 ```math
-E_{\textrm{pair.ext}} = -\frac{1}{2}\int\left(\Delta^{\textrm{(ext)}}(r)\nu^*(r)+\textrm{h.c.}\right)d^3r
+E_{\textrm{pair.ext}} = -\int\left(\Delta^{\textrm{(ext)}}(r)\nu^*(r)+\textrm{h.c.}\right)d^3r
 ```
 * `E_velext`:
 ```math
@@ -60,7 +60,7 @@ $`\vec{j}_{\uparrow}(r) = -\sum_{|E_n|<E_c} \textrm{Im}[u_{n,\uparrow}(r)\nabla
 * `j_b_x`, `j_b_y`, `j_b_z`:  
 $`\vec{j}_{\downarrow}(r) = \sum_{|E_n|<E_c} \textrm{Im}[v_{n,\downarrow}(r)\nabla v_{n,\downarrow}^*(r)]f_{\beta}(-E_n)`$   
 
-In these formulas $`E_{n}`$ denotes quasi-particle energy, and $`E_c`$ is the energy cut-off scale. Fermi distribution function $`f_{\beta}(E)=1/(\exp(\beta E)+1)`$ is introduced to model temperature $`T=1/\beta`$ effects.  
+In these formulas, $`E_{n}`$ denotes quasi-particle energy, and $`E_c`$ is the energy cut-off scale. Fermi distribution function $`f_{\beta}(E)=1/(\exp(\beta E)+1)`$ is introduced to model temperature $`T=1/\beta`$ effects.  
 
 Densities are accessible to users through structure [wslda_density](https://gitlab.fizyka.pw.edu.pl/wtools/wslda/-/tree/public/hpc-engine/wslda_potdens.h):
 ```c
@@ -93,11 +93,11 @@ Minimization of the functional with respect to quasiparticle orbitals provides B
 ```math
 \begin{pmatrix}h_{\uparrow}(r)  &  \Delta(r)+ \Delta_{\textrm{ext}}(r) \\\Delta^*(r)+ \Delta^*_{\textrm{ext}}(r) &  -h^*_{\downarrow}(r)\end{pmatrix} \begin{pmatrix}u_{n\uparrow}(r) \\ v_{n\downarrow}(r)\end{pmatrix}= E_n\begin{pmatrix}u_{n\uparrow}(r) \\ v_{n\downarrow}(r)\end{pmatrix}
 ```   
-where single particle hamiltonian is given by  
+where the single particle Hamiltonian is given by  
 ```math
 h_{\sigma} = -\dfrac{1}{2}\vec{\nabla}\alpha_{\sigma}(r)\vec{\nabla} + V_{\sigma}(r)-\left(\mu_{\sigma}-V_{\sigma}^{\textrm{(ext)}}(r)\right)-\dfrac{i}{2}\left\lbrace \vec{A}_{\sigma}(r)-\vec{v}_{\sigma}^{\textrm{(ext)}}(r),\vec{\nabla} \right\rbrace
 ```   
-Potentials entering the hamiltonian are:  
+Potentials entering the Hamiltonian are:  
 * `alpha_a` and `alpha_b`:  
 $`\alpha_{\sigma} = 2\dfrac{\delta\mathcal{E}_{\textrm{edf}}}{\delta \tau_{\sigma}}`$ -- effective mass,  
 * `V_a` and `V_b`:  
@@ -107,7 +107,7 @@ $`\vec{A}_{\sigma}=\dfrac{\delta\mathcal{E}_{\textrm{edf}}}{\delta \vec{j}_{\sig
 * `delta`:  
 $`\Delta(r)=-\dfrac{\delta\mathcal{E}_{\textrm{edf}}}{\delta \nu^*}`$ -- paring potential. 
 
-Potentials are accessible for user through structure [wslda_potential](https://gitlab.fizyka.pw.edu.pl/wtools/wslda/-/tree/public/hpc-engine/wslda_potdens.h):
+Potentials are accessible for users through the structure [wslda_potential](https://gitlab.fizyka.pw.edu.pl/wtools/wslda/-/tree/public/hpc-engine/wslda_potdens.h):
 ```c
 typedef struct
 {