Moving on a mountain in the direction with the highest rate of change. (The mountain is given by an equation not $z=f(x,y)$). 
A traveler is standing on point $(1,1,0)$ on a mountain that is defined by the equation: 
$3x^2y^3+2x^3y^2z+xyz^3=3$. 
He decides that he wants to move from there in the direction where the rate of change of the height is maximum, in what direction in the space should he move in the start?

I've been faced with these type of questions alot whenever the "mountain" is given by $z=f(x,y)$. And then I know that the direction they asked about would be in the direction of $\nabla f=(f_x,f_y)$ and to decide the third component of the vector, I find the normal to the mountain in the point $(f_x,f_y,-1)$ and make sure $(f_x,f_y,-1)\cdot(f_x,f_y,c)=0$ and I find $c$. 
But here, the equation is making things hard for me, like, is $z$ a function of $x,y$? 
What I thought to do: 
$F(x,y,z)=2x^2y^3+2x^3y^2z+xyz^3=3$, so $\nabla F=(4xy^3, 6x^2y^2,2x^3y^2 )$ (I didn't compute any derivative with $z$ because $z=0$). 
And so that's a normal for the mountain $(4,6,2)$. 
but how do I find the gradient vector of $z=f(x,y)$? if I tried to do $z(2x^3y^2 + xyz^2)=3-2x^2y^3$, I can't really get it into the shape of $z=f(x,y)$. 
Any help is really appreciated in how to deal with this question. 
Thanks in advance.
 A: If you're given an equation of the form $f(x,y,z) = w$, then the partial derivatives of the variables with respect to each other are related by the triple product identity:
$$
\left(\frac{\partial z}{\partial x}\right)_{wy}\left(\frac{\partial w}{\partial z}\right)_{xy}\left(\frac{\partial x}{\partial w}\right)_{yz} = -1,
$$
where the subscripts indicate which variables are being held constant. In this case, we want the partials of $z$ that hold $w$ constant. That is,
$$
\nabla z = \left(\frac{\partial z}{\partial x}\right)_{wy} \hat{\mathbf{x}} + \left(\frac{\partial z}{\partial y}\right)_{wx}\hat{\mathbf{y}} = -\frac{\partial w/\partial x}{\partial w/\partial z} \hat{\mathbf{x}} - \frac{\partial w/\partial y}{\partial w/\partial z} \hat{\mathbf{y}} = -\frac{\partial f/\partial x}{\partial f/\partial z} \hat{\mathbf{x}} - \frac{\partial f/\partial y}{\partial f/\partial z} \hat{\mathbf{y}}.
$$
A: I am going by the equation in the question.
$3x^2y^3+2x^3y^2z+xyz^3 - 3 = 0$
Differentiating with respect to $x$,
$ 6xy^3 + 6x^2y^2z + 2x^3y^2z_x + 3 xyz^2 z_x = 0$
Differentiating with respect to $y$,
$ 9x^2y^2 + 4x^3yz + 2x^3y^2z_y + 3 xyz^2 z_y = 0$
Plug in $(1, 1, 0)$ to find $z_x$ and $z_y$.
