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\begin{document}

\noindent \textbf{Gradi\"enten berekenen}

De gradi\"ent van een potentiaal wordt gegeven door  $\vec{\nabla} V(\vec{r})$:
\begin{description}

\item{a)}  $V(\vec{r})= r^{ \frac{4}{3} }$
\begin{eqnarray*}
\vec{\nabla} r^{ \frac{4}{3} } &=& \frac{4}{3} r^{ \frac{1}{3} } \vec{ \nabla }r \\
			 &=& \frac{4}{3} r^{\frac{1}{3}} \hat{r}
\end{eqnarray*}


\item{b)} $V(\vec{r})= x-3y+7z$

\begin{eqnarray*}
\vec{\nabla}(x-3y+7z) &=& \frac{\partial (x-3y+7z)}{\partial x} \hat{x}+ \frac{\partial (x-3y+7z)}{\partial y} \hat{y}+ \frac{\partial (x-3y+7z)}{\partial z} \hat{z} \\ 
		&=& 1 \hat{x} - 3 \hat{y}+7 \hat{z} \\
\end{eqnarray*}

\item{c)} $V(\vec{r})= x-3y+7z-12$

\begin{eqnarray*}
\vec{\nabla}(x-3y+7z-12) &=& \frac{\partial (x-3y+7z-12)}{\partial x} \hat{x}+ \frac{\partial (x-3y+7z-12)}{\partial y} \hat{y}+ \frac{\partial (x-3y+7z-12)}{\partial z} \hat{z} \\ 
		&=& 1 \hat{x} - 3 \hat{y}+7 \hat{z} \\
\end{eqnarray*}

\item{d)} $V(\vec{r})=e^r$
\begin{eqnarray*}
\vec{\nabla} e^r &=& e^r \vec{\nabla} r \\&=& e^r \hat{r}
\end{eqnarray*}



\end{description}



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