Level Surfaces, Normals and Tangent Planes

A level surface of a function  
  in three dimensional space is the set of points  
  for some constant  

Example: If  
  then the level curves of  
  is the set of parallel planes  
Th normal to a level surface at a point  
\[\mathbf{\nabla} f= (\frac{\partial f}{\partial x} \mathbf{i}+ \frac{\partial f}{\partial y} \mathbf{j}+ \frac{\partial f}{\partial z} \mathbf{k})_{(x_0, y_0, z_0)}\]
  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  
  (sphere centre the origin, radius  
The partial derivatives are  
\[2x, \; 2y, \; 2z\]
  respectively and at  
  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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