Question

Find the gradient of a unit vector normal to level surface

Answer 1

Answer:

Find a normal vector to the surface x3+y3z=3 at the point (1,1,2).

   Solution

   Recall that

   To find a normal vector to a surface, view that surface as a level set of some function g(x,y,z).

   A normal vector to the implicitly defined surface g(x,y,z)=c is ∇g(x,y,z).

   We identify the surface as the level curve of the value c=3 for g(x,y,z)=x3+y3z.

   The gradient of g(x,y,z) is

   ∇g(x,y,z)=3x2 i+3y2z j+y3 k.

   Evaluating at x=1,y=1,z=2, we get

   ∇g(1,1,2)=3 i+6 j+k.

   Hence a normal vector to the surface at (1,1,2) is:

   3 i+6 j+k

Step-by-step explanation: