A surface is determined by the following function :
$x^2 +y^2 +z^4 -z^2 =0 $
The object sits at :$P=(0,\frac{1}{2},-\frac{1}{\sqrt2}) $ on the plane only and it can't fly.
The temperature is determined by : $T(x,y,z)=x-\frac{1}{z} $.
I want to schematically draw the path that the object needs to "walk" in order to reach the maximum temperature available(in every "step" the temperature should increase).
I found that the object needs to start walking in direction of the projection of the $ \nabla T$ on the normal of the plane which is the $ \nabla T$ itself. which is $ \nabla T(p)=(1,0,2)$.
how can i continue would really appreciate some help ?
Giving a surface $S(x,y,z)=0$ and a temperature scalar function $T = T(x,y,z)$ the progress direction is obtained by the projection of $\nabla T = \vec t$ onto the tangent plane to $S$ at the actual point $p_k$ such that $S(p_k)=0$
This projection is given by
$$ \vec u_k = \vec t_k - \hat s_k < \vec t_k, \hat s_k > $$
Here $$ \hat s_k = \frac{\nabla S(p_k)}{||\nabla S(p_k)||} $$
so the next step is obtained by making
$$ p_{k+1}' = p_k + \delta \hat u_k $$
The point $p_{k+1}'$ generally does not pertain to $S$ so after the $p_{k+1}'$ computation an approaching iteration should be executed to obtain $p_{k+1} \in S$
In red the initial point and in blue the path to the top.