Update cheatsheet.en.md

12.temporary_ins/90.electromagnetism-in-vacuum/10.maxwell-equations/20.overview/cheatsheet.en.md

@@ -456,29 +456,29 @@ I recognize here the law of conservation of charge.
   $`\text{div}\,\overrightarrow{j} +\dfrac{\partial \dens^{3D}}{\partial t}=0`$
 --------------------->
 
-* We can *integrate this local equality* over any volume $`\tau`$:
+* We can *integrate this local equality* over any volume $`\tau`$ :   
   <br>
-  $`\displaystyle\iiint_{\tau} \Big(\text{div}\,\overrightarrow{j} +\dfrac{\partial \dens}{\partial t}\big)\,d\tau=0`$
+  $`\displaystyle\iiint_{\tau} \Big(\text{div}\,\overrightarrow{j} +\dfrac{\partial \dens}{\partial t}\big)\,d\tau=0`$     
   <br>
-  *$`\displaystyle\iiint_{\tau} \text{div}\,\overrightarrow{j}\,d\tau+\iiint_{\tau}\dfrac{\partial \dens}{\partial t}\,d\tau=0`$*
+  *$`\displaystyle\iiint_{\tau} \text{div}\,\overrightarrow{j}\,d\tau+\iiint_{\tau}\dfrac{\partial \dens}{\partial t}\,d\tau=0`$*   
 
 * The *Divergence Theorem* (= *Gauss's Theorem*) states that for any vector field
-  $`\overrightarrow{U}`$ and any volume $`\tau`$,
-  *$`\displaystyle\iiint_{\tau} \text{div}\,\overrightarrow{U}\,d\tau=\oiint_S \overrightarrow{U}\cdot dS`$*,
+  $`\overrightarrow{U}`$ and any volume $`\tau`$,   
+  *$`\displaystyle\iiint_{\tau} \text{div}\,\overrightarrow{U}\,d\tau=\oiint_S \overrightarrow{U}\cdot dS`$*,   
   $`S`$ being the closed surface bounding the volume $`\tau`$.   
   <br>
-  *Applied to the first term* of the equality, we obtain :
+  *Applied to the first term* of the equality, we obtain :   
   <br>
-  **$`\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} + \iiint_{\tau}\dfrac{\partial \dens}{\partial t}\,d\tau=0`$** .
+  **$`\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} + \iiint_{\tau}\dfrac{\partial \dens}{\partial t}\,d\tau=0`$**.   
 
-* Noting again that *space and time are independent in classical physics*, the order of derivation or integration with respect to a spatial variable and a time variable does not matter:
+* Noting again that *space and time are independent in classical physics*, the order of derivation or integration with respect to a spatial variable and a time variable does not matter :   
   <br>
-  **$`\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} +\dfrac{\partial}{\partial t} \left(\iiint_{\tau}\dens \,d\tau\right)=0`** .
+  **$`\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} +\dfrac{\partial}{\partial t} \left(\iiint_{\tau}\dens \,d\tau\right)=0`$**.   
 
-* Noting that *$`\displaystyle\iiint_{\tau}\dens^{3D} \,d\tau`$ is the total charge $`Q_{int}`*
-  contained in the volume $`\tau`$, we obtain the **integral expression of the conservation law** of charge:
+* Noting that *$`\displaystyle\iiint_{\tau}\dens^{3D} \,d\tau`$ is the total charge $`Q_{int}`$*
+  contained in the volume $`\tau`$, we obtain the **integral expression of the conservation law** of charge :   
   <br>
-  **$`\mathbf{\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} +\dfrac{dQ_{int}}{dt}=0}`** .
+  **$`\mathbf{\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} +\dfrac{dQ_{int}}{dt}=0}`$**.   
 
 <br>
 
@@ -493,19 +493,19 @@ I recognize here the law of conservation of charge.
 * The **sensitivity** of a particle **to electromagnetic interaction** is quantified
   by the parameter called *electric charge* of the particle.
 
-* The force that describes the *action of an electromagnetic field $`\big(\overrightarrow{E}, \overrightarrow{B}\big)`*
-  on a particle with charge $`q`$ is the **Lorentz force**, with the expression:
+* The force that describes the *action of an electromagnetic field $`\big(\overrightarrow{E}, \overrightarrow{B}\big)`$*
+  on a particle with charge $`q`$ is the **Lorentz force**, with the expression :   
   <br>
   **$`\overrightarrow{F}_{Lorentz}=q\Big(\overrightarrow{E}+\overrightarrow{v}\times\overrightarrow{B}\Big)`$**
   <br>
-  &nbsp;&nbsp;where $`\overrightarrow{v}`$ is the velocity vector of the particle in the inertial frame of the observer.
+  &nbsp;&nbsp;where $`\overrightarrow{v}`$ is the velocity vector of the particle in the inertial frame of the observer.   
 
-* *During an elementary displacement $`\overrightarrow{dl}`* of the particle in the electromagnetic field
-  $`\big(\overrightarrow{E}, \overrightarrow{B}\big)`*, the **work of the Lorentz force** is written as:
+* *During an elementary displacement $`\overrightarrow{dl}`$* of the particle in the electromagnetic field
+  $`\big(\overrightarrow{E}, \overrightarrow{B}\big)`$*, the **work of the Lorentz force** is written as :   
   <br>
-  **$`d\mathcal{W}_{Lorentz}=\overrightarrow{F}_{Lorentz}\cdot\overrightarrow{dl}`**,
+  **$`d\mathcal{W}_{Lorentz}=\overrightarrow{F}_{Lorentz}\cdot\overrightarrow{dl}`**,   
   <br>
-  that is,
+  that is,   
   <br>
   $`\begin{align}
   d\mathcal{W}_{Lorentz}&=q\Big(\overrightarrow{E}+\overrightarrow{v}\times\overrightarrow{B}\Big)\cdot\overrightarrow{dl}\\
@@ -513,14 +513,14 @@ I recognize here the law of conservation of charge.
   &= q\Big(\overrightarrow{E}\cdot\overrightarrow{dl}\Big)+ q\Big(\overrightarrow{v}\times\overrightarrow{B}\cdot\overrightarrow{dl}\Big)\\
   &\\
   &= q\,\overrightarrow{E}\cdot\overrightarrow{dl} + q\Big(\overrightarrow{v},\overrightarrow{B},\overrightarrow{dl}\Big) \\
-  \end{align}`$
+  \end{align}`$   
   <br>
-  where $`\Big(\overrightarrow{v},\overrightarrow{B},\overrightarrow{dl}\Big)`$ is the scalar triple product of the sequence of the three vectors.
+  where $`\Big(\overrightarrow{v},\overrightarrow{B},\overrightarrow{dl}\Big)`$ is the scalar triple product of the sequence of the three vectors.   
 
-* Since the *vectors $`\overrightarrow{v}`$ and $`\overrightarrow{dl}=\overrightarrow{v}\,dt`* are *collinear*, the scalar triple product
-  is zero:
+* Since the *vectors $`\overrightarrow{v}`$ and $`\overrightarrow{dl}=\overrightarrow{v}\,dt`$* are *collinear*, the scalar triple product
+  is zero :   
   <br>
-  *$`\Big(\overrightarrow{v},\overrightarrow{B},\overrightarrow{dl}\Big)=0`$*,
+  *$`\Big(\overrightarrow{v},\overrightarrow{B},\overrightarrow{dl}\Big)=0`$*,   
 
 !!!!
 !!!! <details markdown=1>