Help with the gradient operator

If i need to describe how the direction of grad(f) (for a specific f) varies in space by considering grad(f) on surfaces given by f(x,y,z)=constant, how do I do this? What is the question asking? What relevence does grad(f) on a surface f(x,y,z)=constant have?

Any hints or tips appreciated

Thanks

Re: Help with the gradient operator

Quote:

Originally Posted by

**darren86** If i need to describe how the direction of grad(f) (for a specific f) varies in space by considering grad(f) on surfaces given by f(x,y,z)=constant, how do I do this? What is the question asking? What relevence does grad(f) on a surface f(x,y,z)=constant have?

$\displaystyle \nabla f\left( {x_0 ,y_0 ,z_0 } \right)$ is the normal of the tangent plane to the surface $\displaystyle f(x,y,z)=C$ at the point $\displaystyle (x_0 ,y_0 ,z_0)$.

Re: Help with the gradient operator

Quote:

Originally Posted by

**Plato** $\displaystyle \nabla f\left( {x_0 ,y_0 ,z_0 } \right)$ is the normal of the tangent plane to the surface $\displaystyle f(x,y,z)=C$ at the point $\displaystyle (x_0 ,y_0 ,z_0)$.

Thanks for your reply. If f(x,y,z) = x^2+y^2 for instance what can be said about how the direction of grad(f) varies in space (by considering surfaces where f=constant)?

I don't think it is sufficient to just say that for points on any surface f=constant, gradf will be normal to the surface. What else can be said?

Re: Help with the gradient operator

Quote:

Originally Posted by

**darren86** I don't think it is sufficient to just say that for points on any surface f=constant, gradf will be normal to the surface. What else can be said?

I don't really understand what you are attempting to do.

We can also say that $\displaystyle \nabla f$ points in the direction of maximum increase in the field and $\displaystyle \|\nabla f\|$ is the rate of maximum increase.