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\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
{Integral Definitions of Gradient, Divergence and 
Curl} (February 11, 2013) \\
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\medskip

\noindent The case of rectangular volume
\begin{eqnarray} 
  V = \lim_{V\to 0} \int_V d\tau = 
  \lim_{V\to 0} \int_V dx\,dy\,dz 
\end{eqnarray} 
is considered. $S$ is the surface of $V$.
\medskip

\noindent Gradient (Book p.66):
\begin{eqnarray} 
  &~& \nabla\phi\ =\ 
  \lim_{V\to 0} \frac{\int_S\phi d\vec{\sigma}}{\int_Vd\tau} 
  \\ \nonumber &=& \left[\phi(x,y,z+dz)-\phi(x,y,z)\right]\,
                   \frac{dx\,dy\,\hat{z}}{dx\,dy\,dz}
  \\ \nonumber &+& \left[\phi(x+dx,y,z)-\phi(x,y,z)\right]\,
                   \frac{dy\,dz\,\hat{x}}{dx\,dy\,dz}
  \\ \nonumber &+& \left[\phi(x,y+dy,z)-\phi(x,y,z)\right]\,
                   \frac{dz\,dx\,\hat{y}}{dx\,dy\,dz}
  \\ \nonumber &=& \frac{\partial\phi}{\partial z}\,dz\,
                   \frac{dx\,dy\,\hat{z}}{dx\,dy\,dz}
                +  \frac{\partial\phi}{\partial x}\,dx\,
                   \frac{dy\,dz\,\hat{x}}{dx\,dy\,dz}
  \\ \nonumber &+& \frac{\partial\phi}{\partial y}\,dy\,
                   \frac{dz\,dx\,\hat{y}}{dx\,dy\,dz}
                =  \frac{\partial\phi}{\partial z}\,\hat{z}
                +  \frac{\partial\phi}{\partial x}\,\hat{x}
                +  \frac{\partial\phi}{\partial y}\,\hat{y}\,.
\end{eqnarray} 

\noindent Divergence (Book p.66):
\begin{eqnarray} 
  &~& \nabla\cdot\vec{A}\ =\ \lim_{V\to 0} 
      \frac{\int_S\vec{A}\cdot d\vec{\sigma}}{\int_Vd\tau} 
  \\ \nonumber &=& \left[A_z(x,y,z+dz)-A_z(x,y,z)\right]\,
                   \frac{dx\,dy}{dx\,dy\,dz}
  \\ \nonumber &+& \left[A_x(x+dx,y,z)-A_x(x,y,z)\right]\,
                   \frac{dy\,dz}{dx\,dy\,dz}
  \\ \nonumber &+& \left[A_y(x,y+dy,z)-A_y(x,y,z)\right]\,
                   \frac{dz\,dx}{dx\,dy\,dz}
  \\ \nonumber &=& \frac{\partial A_z}{\partial z}\,dz\,\frac{1}{dz}
                +  \frac{\partial A_x}{\partial x}\,dx\,\frac{1}{dx}
                +  \frac{\partial A_y}{\partial y}\,dy\,\frac{1}{dy}
  \\ \nonumber &=& \frac{\partial A_z}{\partial z}
  +\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}\,.
\end{eqnarray} 

\noindent Curl (Book p.66):
\begin{eqnarray} 
  &~& \nabla\times\vec{A}\ =\ \lim_{V\to 0} 
      \frac{\int_S d\vec{\sigma}\times\vec{A}}{\int_Vd\tau} 
\end{eqnarray} 
\begin{eqnarray} \nonumber
  &~& d\vec{\sigma}\times\vec{A}\ = \\ \nonumber 
  &~& dx\,dy\,\hat{z}\times\left[\vec{A}(x,y,z+dz)-\vec{A}(x,y,z)\right]
  \\ \nonumber 
  &+& dy\,dz\,\hat{x}\times\left[\vec{A}(x+dx,y,z)-\vec{A}(x,y,z)\right]
  \\ \nonumber 
  &+& dz\,dx\,\hat{y}\times\left[\vec{A}(x,y+dy,z)-\vec{A}(x,y,z)\right]
  \\ \nonumber 
  &=& dx\,dy\,\hat{z}\times\frac{\partial\vec{A}}{\partial z}\,dz
   +  dy\,dz\,\hat{x}\times\frac{\partial\vec{A}}{\partial x}\,dx
  \\ \nonumber 
  &+& dz\,dx\,\hat{y}\times\frac{\partial\vec{A}}{\partial y}\,dy
\end{eqnarray} 
\begin{eqnarray} \nonumber
  &~& \frac{d\vec{\sigma}\times\vec{A}}{dx\,dy\,dz} 
   =  \hat{z}\times\frac{\partial\vec{A}}{\partial z}
   +  \hat{x}\times\frac{\partial\vec{A}}{\partial x}
   +  \hat{y}\times\frac{\partial\vec{A}}{\partial y}\\ \nonumber 
  &=& \hat{y}\,\frac{\partial A_x}{\partial\,z} 
   -  \hat{x}\,\frac{\partial A_y}{\partial\,z} 
   +  \hat{z}\,\frac{\partial A_y}{\partial\,x} 
   -  \hat{y}\,\frac{\partial A_z}{\partial\,x} \\ \nonumber
  &+& \hat{x}\,\frac{\partial A_z}{\partial\,y} 
   -  \hat{z}\,\frac{\partial A_x}{\partial\,y} \\ \nonumber
  &=& \hat{x}\,\left(\frac{\partial A_z}{\partial\,y} -
                     \frac{\partial A_y}{\partial\,z}\right)
   +  \hat{y}\,\left(\frac{\partial A_x}{\partial\,z} -
      \frac{\partial A_z}{\partial\,x}\right) \\ \nonumber
  &+& \hat{z}\,\left(\frac{\partial A_y}{\partial\,x} -
                     \frac{\partial A_x}{\partial\,y}\right)
\end{eqnarray} 

\noindent Curl with Levi-Civita:
\begin{eqnarray} 
   \nabla\times\vec{A}\ =\ \lim_{V\to 0} \frac{\int_S 
   \epsilon_{ijk}\,\hat{x}_i\,d\sigma_j\,A_k}{\int_Vd\tau} 
\end{eqnarray} 
with
$$ d\sigma_1 = dx_2\,dx_3,~ d\sigma_2 = dx_3\,dx_1,~ 
   d\sigma_3 = dx_1\,dx_2.                             $$
\begin{eqnarray} \nonumber
   &~&\epsilon_{ijk}\,\hat{x}_i\,d\sigma_j\,A_k\ = \\ \nonumber 
   &~& \hat{x}_1\,\left(d\sigma_2\,\partial_2A_3\,dx_2
       - d\sigma_3\,\partial_3A_2\,dx_3\right) \\ \nonumber
   &+& \hat{x}_2\,\left(d\sigma_3\,\partial_3A_1\,dx_3
       - d\sigma_1\,\partial_1A_3\,dx_1\right) \\ \nonumber
   &+& \hat{x}_3\,\left(d\sigma_1\,\partial_1A_2\,dx_1
       - d\sigma_2\,\partial_2A_1\,dx_2\right) \\ \nonumber
\end{eqnarray} 
\begin{eqnarray} \nonumber
   &~& \epsilon_{ijk}\,\frac{\hat{x}_i\,d\sigma_j\,A_k}{dx\,dy\,dz} 
   \ = \\ \nonumber &~&\hat{x}_1\,(\partial_2A_3-\partial_3A_2) 
                 + \hat{x}_2\,(\partial_3A_1-\partial_1A_3) 
   \\ \nonumber &+&\hat{x}_3\,(\partial_1A_2-\partial_2A_1)\ =\
      \epsilon_{ijk}\,\hat{x}_i\,\partial_j\,A_k\,.
\end{eqnarray} 

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