Hey blaisem.
This is known as the total derivative and is based on the chain rule in multiple dimensions:
Total derivative - Wikipedia, the free encyclopedia
Hello, please excuse my title if I stated the topic incorrectly.
I was given an assignment to derive the quantum mechanical operator for the z-component of the angular momentum in spherical coordinates. I have found the solution, and the derivation uses the following relationship:
= + +
I was curious if anyone might be able to tell me from what this relationship is derived from. Whenever I search "transformation to spherical coordinates" or something along those lines, I find explanations to transforming each cartesian coordinate into spherical representation, but I don't see any transformation for the partial derivative of a cartesian coordinate into spherical representation.
If anyone could help me with the "correct" term for the above relationship or the mathematical technique used to derive it, I will gladly google the details then myself.
Thank you very much!
Hey blaisem.
This is known as the total derivative and is based on the chain rule in multiple dimensions:
Total derivative - Wikipedia, the free encyclopedia
Hi chiro,
Thanks a lot for the information! I have since found a couple other resources that have helped me in this process for anyone else interested:
Wolfram Demonstrations Project
Envisioning total derivatives of scalar functions f(x,y) ©
I have a couple of questions after looking at this to help complete my understanding, if you or anyone else would be willing to take a stab at them I'd appreciate it! I'll start by trying to outline my understanding of the topic currently, then I'll summarize my questions at the end.
So, if I have understood this correctly, the relationship above means that the rate of change of y is given by three "vector components" corresponding to r, theta, and phi. The rate of change of y is then described by the rate of change of each variable multiplied respectively by the distance of the variable. If this is correct, then I know why the total derivative is formulated the way it is.
Now, the total derivative is only used when the variables are not independent. My question would then be: in the relationship I have in my original post, why would the radius, theta, and phi necessarily be dependent on one another? Is it because they can all change at the same time, ie. as each variable is varied, the others are not necessarily held constant? The problem with this explanation is that if I try to extend it to a general case, it seems to conflict with my understanding of partial derivatives:
Taking the partial derivative with respect to one variable only describes how that variable changes when the rest are held constant. What exactly does one accomplish then when one takes the partial derivative of a function f(x,y)? Since the partial derivative of both variables are taken, then both variables are allowed to change (ie are not held constant). This is exactly what I am doing in my relationship from the original post. So I am currently stumbling on:
1) What is the purpose of taking the partial derivative of all variables vs. the total derivative? My understanding of each seems to have their purposes overlapping.
2) Why are the variables in my relationship considered dependent on one another, and therefore characterized appropriately by the total derivative instead of partial derivatives? If I were to approach the problem of deriving the relationship above without having ever seen it before, I would have not have been able to tell you whether the spherical coordinates should be handled as dependent on one another or independent.
I hope I adequately explained where I am coming from for my issue. Thanks for any advice again!
For 1) it depends on what you are doing but the reason for the total derivative is that you want to find the overall vector derivative that takes into account all of the individual components (like x,y or r,theta etc).
A normal function maps R^n -> R and the total derivative tells us the total change by considering the vectors.
Think of it like Pythagoras' theorem where the length of the hypotenuse is the sum of the squares of the rest of the sides (even in n-dimensions which this extends to in Euclidean/Cartesian geometry).
For 2) it depends on the nature of the function and the geometry but essentially we look at the smallest number of independent components and then consider these in the context of the chain rule where we consider transformations that take us to the atomic simplest independent variables.
Basically as an example consider u(v(w(x))) where x is the atomic variable and the composition functions are chained together where the final derivative is in terms of du/dv * dv/dw * dw/dx.
As for d/d(xy), you can only do this if xy is a single variable.
So think of it in terms of of Pythagoras' theorem but instead of lengths of some solid rectangle, they are rates of change and behave according to the laws of differentiation.
In fact we generalize geometry in the exact same way by modelling ds^2 where ds is the rate of change of length of a vector in a generic co-ordinate system and this is typically referred to as tensor analysis or Riemannian geometry or differential geometry.