tangent plane to level set

Question

I am confused betweeen tangent plane to the level set(is it the same as level surface?) and to the tangent plane on the surface?

I know the formula $z=f(x,y): z-z_0 = f_{x}(x_0)(x-x_0) +f_y(y_0)(y-y_0)$ but cannot understand for which case it corresponds. Also I found some example where it was just without $z$ and $z_0$ : $f_{x}(x_0)(x-x_0) +f_y(y_0)(y-y_0)=0$

What whould be the difference between two tangent planes? It would be very nice if someone could explain it on the numerical example.

Answer 1

This one

$$z-z_0 = f_{x}(x_0)(x-x_0) +f_y(y_0)(y-y_0)$$

is the general expression for the tangent plane to $z=f(x,y)$ at the point $(x_0,y_0,z_0)$ while

$$f_{x}(x_0)(x-x_0) +f_y(y_0)(y-y_0)=0$$

is the intersection of the tangent plane with $x-y$ plane $(z=0)$ when $z_0=0$.

For example $z=f(x,y)=x^2+y^2 \implies f_x=2x \quad f_y=2y$ then at $(1,1,2)$ the tangent plane is

$$z-2=2(x-1)+2(y-1)$$

Answer 2

Case $1)$ $$z= f(x,y)$$

Tangent plane is : $$z-z_0 = f_{x}(x_0)(x-x_0) +f_y(y_0)(y-y_0)$$

Case $2)$

$$ F(x,y,z)=0$$

The normal to the tangent plane is $$N=\nabla F =(\partial F/\partial x,\partial F/\partial y,\partial F/\partial z)$$ evaluated at $(x_0,y_0,z_0).$

The equation of Tangent plane is then,

$$ a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$

where $N=(a,b,c)$ is the said normal vector.

Note that Case $2$ is a generalization of case $1$

We can wright $z= f(x,y)$ as $$ F(x,y,z)= z- f(x,y)=0$$