# Help with the gradient operator

• November 17th 2012, 09:09 AM
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?

Any hints or tips appreciated

Thanks
• November 17th 2012, 10:00 AM
Plato
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?

$\nabla f\left( {x_0 ,y_0 ,z_0 } \right)$ is the normal of the tangent plane to the surface $f(x,y,z)=C$ at the point $(x_0 ,y_0 ,z_0)$.
• November 18th 2012, 06:24 AM
darren86
Re: Help with the gradient operator
Quote:

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

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?
• November 18th 2012, 08:16 AM
Plato
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 $\nabla f$ points in the direction of maximum increase in the field and $\|\nabla f\|$ is the rate of maximum increase.