Update 3_drude_model.md - fix

src/3_drude_model.md

@@ -282,7 +282,7 @@ We apply a magnetic field $\bf B$ along the $z$-direction to a planar (two-dimen
   1. Suppose we measure a Hall voltage $V_H$. Express the Hall resistance $R_{xy} = V_H/I$ as a function of magnetic field. Does $R_{xy}$ depend on the geometry of the sample? Also express $R_{xy}$ in terms of the Hall coefficient $R_H$.
 
 ??? question "What is the relation between the electric field and the electric potential?"
-  $V_b - V_a = -\oint_{\Gamma} \mathbf{E} \cdot d\mathbf{\ell}$ if $\Gamma$ is a path from $a$ to $b$.
+  $V_b - V_a = -\int_{\Gamma} \mathbf{E} \cdot d\mathbf{\ell}$ if $\Gamma$ is a path from $a$ to $b$.
   
   2. Assuming we control the magnetic field $\mathbf{B}$, what quantity can we extract from a measurement of the Hall resistance? Would a large or a small magnetic field give a Hall voltage that is easier to measure?
   3. Express the longitudinal resistance $R=V/I$, where $V$ is the voltage difference over the sample along the $x$ direction, in terms of the longitudinal resistivity $ρ_{xx}$. Suppose we extracted $n$ from a measurement of the Hall resistance, what quantity can we extract from a measurement of the longitudinal resistance? Does the result depend on the geometry of the sample?
@@ -314,8 +314,8 @@ $$\mathbf{E} = \begin{pmatrix} ρ_{xx} & ρ_{xy} \\ ρ_{yx} & ρ_{yy} \end{pmatr
   1. Sketch $ρ_{xx}$ and $ρ_{xy}$ as a function of the magnetic field $\bf{B}$.
   2. Invert the resistivity matrix to obtain the conductivity matrix,
 
-$$ \begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix}$$
+     $$ \begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix}$$
 
-allowing you to express $\mathbf{j}$ as a function of $\mathbf{E}$.
+     allowing you to express $\mathbf{j}$ as a function of $\mathbf{E}$.
   3. Sketch $\sigma_{xx}$ and $\sigma_{xy}$ as a function of the magnetic field $\bf B$.
   4. Give the definition of the Hall coefficient. What does the sign of the Hall coefficient indicate?