Tangent Plane and Normal Vector

The gradient vector field of a function

f |\x,y,z/|

is defined by

@f- @f- @f-- \~/ f = @x i+ @y j+ @z k.

At a point

|\a,b,c/|

the gradient vector is normal to the level surface

f|\x,y,z/| = f|\a,b,c/|

containing the point and determines the orientation of the plane tangent to the level surface.

Below is the graph of part of the level surface

x2 + y2 + z2 = 1

of the function

f |\x,y,z/| = x2 + y2 + z2

whose gradient vector is

\~/ f = 2xi+ 2y j+ 2zk.

At the point

( V~ -) 1 1 2 -,-, --- 2 2 2

the gradient vector becomes

V~ -- \~/ f = i+ j+ 2 k

and the equation of the tangent plane is

x + y + V~ 2z = 2.