Level Surfaces, Normals and Tangent Planes

A level surface of a function
$f(x,y,z)$
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
$x^2+y^2+z^2=14$
$\sqrt{14}$
).
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
$2(x-1)+4(y-2)+6(z-3)=0 \rightarrow x+2y+3z=14$
.

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