Level Surfaces, Normals and Tangent Planes
\[f(x,y,z)\]
in three dimensional space is the set of points \[(x,y,z)\]
satisfying \[f(x,y,z)=c\]
for some constant \[c\]
Example: If
\[f(x,y,z)=x+3y+2z\]
then the level curves of \[f\]
is the set of parallel planes \[x+3y+2z=c\]
.Th normal to a level surface at a point
\[(x_0,y_0,z_0)\]
is \[\mathbf{\nabla} f= (\frac{\partial f}{\partial x} \mathbf{i}+ \frac{\partial f}{\partial y} \mathbf{j}+ \frac{\partial f}{\partial z} \mathbf{k})_{(x_0, y_0, z_0)}\]
and the tangent plane is \[\frac{ \partial f}{\partial x}|_{(x_0,y_0,z_0)} (x-x_0) + \frac{ \partial f}{\partial y}|_{(x_0,y_0,z_0)}(y-y_0) + \frac{ \partial f}{\partial z}|_{(x_0,y_0,z_0)}(z-z_0)=0\]
.Example: For the level surface
\[x^2+y^2+z^2=14\]
(sphere centre the origin, radius \[\sqrt{14}\]
).The partial derivatives are
\[2x, \; 2y, \; 2z\]
respectively and at \[(1,2,3)\]
these take the values 2, 4 and 6.The normal is
\[2 \mathbf{u}+ 4 \mathbf{j} + 6 \mathbf{k}\]
and the tangent plane is \[2(x-1)+4(y-2)+6(z-3)=0 \rightarrow x+2y+3z=14\]
. 
