I have a specific smooth function $F(x,y,z) = 0$ which implicitly defines a function $\widehat{z}(x,y)$.
In my analysis of this function's properties (details not too relevant), the term $$M(x,y) \equiv \frac{\widehat{z}_y}{\widehat{z}_x}$$ (that is, the ratio of partial derivatives of $\widehat z$.) appears frequently.
I'm wondering if there is geometric intuition for what $\widehat{z}_y/\widehat{z}_x$ represents; a visual aid or reference to known other occurrences of this term would be helpful.
From what I understand, at any particular point $p$ on the surface, $\widehat{z}_x$ and $\widehat{z}_y$ define the tangent plane at $p$. And $M$ seems like the slope of a certain line. But beyond this, e.g. if the line has something to do with level curves etc., I am not sure.
$\frac{\hat z_y}{\hat z_x}$ is directly related to the direction of the level curve at the given point.
Let's assume that equation $F(x,y,z)=0$ also defines function $\hat x(y,z)$, at least locally.
We have, for any $y,z$: $$ \hat z(\hat x(y,z),y) = z$$ Let's calculate the full derivative of this equality over $y$, using the chain rule: $$ \hat z_x\big(\hat x(y,z),y\big)\hat x_y(y,z) + \hat z_y\big(\hat x(y,z),y\big)= 0 $$ so $$ \frac{\hat z_y\big(\hat x(y,z),y\big)}{\hat z_x\big(\hat x(y,z),y\big)} = - \hat x_y(y,z)$$ $$ \frac{\hat z_y(x,y)}{\hat z_x(x,y)} = - \hat x_y(y,\hat z(x,y))$$ $\hat x_y(y_0,z_0)$ is the tangens of the angle that the line tangent to the curve $(\hat x(y,z_0),y,z_0)$ at point $(\hat x(y_0,z_0),y_0,z_0)$ makes with the plane $yz$. This curve is the level curve that can be obtained by taking the interesction of surface given by the equation $F(x,y,z)=0$ with the plane $z=z_0$.
That means that if you take the surface given by $F(x,y,z)=0$, intersect it with a plane $z= \hat z(x_0,y_0)$ obtaining the level curve, and then consider a line tangent to this level curve at point $(x_0,y_0,\hat z(x_0,y_0))$, then the tangens of the angle it makes with the plane $yz$ will be equal to $-\frac{\hat z_y(x_0,y_0)}{\hat z_x(x_0,y_0)}$.