In this post, I just want to motivate why a gradient gets complicated in a non-euclidean space, and in a following post I’ll work through how to take a gradient. The reason we care is because to find the force from potential energy, we need to take a gradient.
In a euclidean space with cartesian coordinates, the gradient can be taken this way
Grad(U) = (∂U/∂x) x-hat + (∂U/∂y) y-hat + (∂U/∂z) z-hat
One way of interpreting the operation ∂/∂x is that it “nudges” whatever the scalar function along in the direction of x, and then looks at how that things change. It also keeps track of how far the nudging has occured, and it takes a ratio.
This “nudging” is really important to get a handle on. See, in a euclidean geometry described by cartesian coordinates, a nudge in the x-direction is always the same nudge no matter what; and a nudge in the x-direction never ever causes a nudge in the y-direction or the z-direction. That’s actually what this metric
ds² = dx² + dy² + dz²
tells you. The 1 in front of dx² tells you that the dot product dx.dx is a constant of unity. The fact that there is a (hidden) zero in front of dx.dy and dx.dz tells you that nudges in x are not influenced by or influence nudges in y or z directions.
But if you remember back to my first post, we had this metric:
ds² = (1+α²) dx² + (1+β²) dy² + αβ dx dy
In this metric, we have to three things to consider when nudging:
(1) A nudge in the x-direction is not always the same nudge depending on where you are, unless α is a constant; and similarly for the y-direction.
(2) A nudge in the x-directions isn’t the same nudge as a nudge in the y-direction, unless α = β.
(3) A nudge in the x direction will co-occur with a nudge in the y-direction as along, unless both α and β are not equal to zero
Another way of thinking about this is like this, imagine you had the vector x-hat, which is a unit vector. Imagine you tried to drag x-hat in the x-direction. In a euclidean space, x-hat would still have a magnitude of 1, and x-hat would still be facing in the x-direction. However, in a non-euclidean geometry, two things can happen. One kind of that can happen is that x-hat can grow or shrink. That growing or shrinking is caused by terms like α² or β². Another kind of thing that can happen is that you’re x-hat vector, which you dragged in the x-direction, is all of sudden pointing a little bit in another direction. That’s what the term αβ does. It twists your unit vectors around.
This growing, shrinking, and twisting is what we’re going to have to keep track of in order to take a gradient. How we keep track of this is the subject for another post, but I hope you get the big picture–a nudge is not just a nudge but a nudge with a growth and a twist.