A valley is described by $h(r)=e^{xy}$ w/ $r=(x,y)^T$. With the gradient $\nabla h_r=(\partial_xh,\partial_yh)^T=e^{xy}*(y,x)^T$ and the total differential $dh_r(n)=\partial_xh*n_x+\partial_yh*n_y=e^{xy}*(xn_x+yn_y)$
I now need to find the direction of the gradient vector and the contour lines of said gradient.
I thought of using a parallel unit vector to the gradient for it's direction $$n_{||}=\frac{\nabla h_r}{||\nabla h_r||}$$
And then using a vector ($n_{\bot}$) perpendicular to that, to get the direction of the contour lines.
The problem I am facing is that I just can't seem to be able to calculate them explicitly.