I'm not too sure how to sure that the vector field would be orthogal to the graph of f when f is constant mathematically -how ever I do know the vector field points towards the direction of maximal increase, can this explain orthogonality?
Also, how can I indicate a maximum or a minimum on a gradient vector field?
May 14th 2012, 03:35 PM
Re: Vector Fields
I don't understand what you mean by "f is constant". If y= f(x)= C, the graph is a horizontal line in the xy-plane and vectors normal to it are of the form <0, a> for any number, a. If z= f(x,y)= C, the graph is a plane parallel to the xy-plane and vectors normal to it are of the form <0, 0, a> for any number, a.
vectors do not form a linearly ordered set. It makes no sense to talk about maximum or minimum vectors. You can talk about maximum or minimum length of a set of vectors. is that what you meant?