I hope this is a relatively intelligent question (keeping in mind that I'm not a mathematician),...
I have an error surface and I'm trying to find an equation that will give me the slope at a given point, in a given direction. I know this is effectively the directional directive, but the directional directive seems to always be defined according to the partial derivatives and I have a case where there is 'something interesting happening' that is not reflected in the partial derivatives (see my previous posting to get an idea of my motivation).
I have found this equation for 3 dimensional space and everything seems to work well,... I basically just took the equation for the error, rotated it by an angle 'theta', translated my 'current position' on this surface to the origin and took the derivative relative to x. With this equation I am able to generate graphs that show me how the error/slope changes as I rotate on a given point on my error surface. Now I'm wondering if this can also be done for n-dimensional spaces. In particular I'm wondering how many of my 'intuitions' can be extended to higher dimensional spaces. Specifically:
In 3 dimensions, if I rotate about any axis, the other axes eventually resolve into each other,... For example if I rotate about the z-axis (the error), then eventually what was the x-axis will become the y-axis and vice-versa. But does this also happen when I have more than 3 dimensions? If for example I have x, y, z and w dimensions and I rotate around the z-axis, will the x-axis eventually become the y and w-axis?
In 3 dimensions, a 360 degree sweep of theta is sufficient to rotate through all the axes (in this case just the two: x -> y). Is this notion also applicable to higher dimensions? Will I also be able to rotate an angle 360 degrees and sweep through all axes except the one that I'm rotating around?
Thanks for your patience and please be gentle :)
Terry