Let f and g be functions R^3 -> R

[Pearce_09]

the question is
Let f and g be functions R^3 -> R. suppose f is differentiable and

df(x,y,z) = ( df/dx , df/dy , df/dz ) = g(x,y,z)(x,y,z)
show that f is constant on any sphere of radius r centered at the origin defined by x^2 + y^2 + z^2 = r^2.

this means to show that F is the level curves S: x^2 + y^2 + z^2 = r^2. You do this by showing that the gradient of f is normal to f; therefore the inner (dot) product of the gradient of f and the tangential vector is equal to 0

But i have no idea how to do it. Please help
thanks
A.P.

 

[CarlB]

You do this by showing that the gradient of f is normal to f; therefore the inner (dot) product of the gradient of f and the tangential vector is equal to 0

If you try to do it with tangent vectors, and somehow manage to make your way to a solution, you may end up feeling really silly later on.

Think about how to use the dot product of r=(x,y,z) with df to show that df is radial.

Carl