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===== Lab 11: A gentle rain falls on Monte Sol =====

The function
\begin{eqnarray*}
f(x,y) = 2 e^{-(x-1)^2 - (y-1)^2} + e^{-(x+1)^2 - y^2}
\end{eqnarray*}
is a rough scale model of a pair of small mountains named Monte Sol and Monte Luna on the outskirts of Santa Fe, New Mexico. Monte Sol, the bigger of the two mountains is about 200 meters high above the plain, so the scale is 1 = 100 meters. 

In this lab you will use vector calculus, 3D graphics, and numerical integration of differential equations to explore how water flows when a gentle rain falls on the mountains.


**Problem 1.** Make a surface plot of Monte Sol and Monte Luna. It should look something like this.

{{:gibson:teaching:spring-2015:math445:montesol.png?direct&400|}}


**Problem 2.** In a gentle rainstorm, water will flow down the mountains in the direction of steepest descent, i.e. along the negative of the gradient of $f$. Find the gradient of $f$ using elementary calculus, then make an $x,y$ plot with both contours of the mountain height $f$ and a quiver plot showing the direction of flow of rainwater. It should look something like this. 

{{:gibson:teaching:spring-2015:math445:montesol_quiver.png?direct&400|}}

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