- #1

Runei

- 193

- 17

I was working on a problem using Euler-Lagrange equations, and I started wondering about the total and partial derivatives. After some fiddling around in equations, I feel like I have confused myself a bit.

I'm not a mathematician by training, so there must exist some terminology which could help explain my problem in a more clear sense, but alas, it is unkown to me :) But here goes.

Suppose we have a function ##f(x,y,z,t)## where the variables are themselves through composition, functions of t.

$$x \rightarrow x(y,z,t)\\

y \rightarrow y(x,z,t)\\

z \rightarrow z(x,y,t)$$

If the functions for x, y and z trivially related to each other, we can back-substitute and get the function ##f## to be just a function of ##t##.

Suppose I wanted to find the derivative ##df/dt##, I could start by doing the total derivative of ##f##.

$$\dfrac{df}{dt} = \sum_i \dfrac{\partial{f}}{\partial{\sigma_i}}\dfrac{d \sigma_i}{d t}$$

Where ##\sigma_i## represents one of the four parameters ##(x,y,z,t)##.

My first insight (I hope), is that the second derivative ##d\sigma_i/dt## is the corresponding

*total derivative*of the corresponding variable (x,y,z,t).

If the relationships between x, y, z and t are such that I can easily back-substitute the functions and generate a single function of just t. For example if ##x\rightarrow x(y,z,t)##, ##y\rightarrow y(z,t)## and ##z \rightarrow z(t)##, then I can substitute them all back into ##f## until there is only #t# left. Also, the total derivative will work beautifully as follows (using subscript notation for partial derivatives):

$$\dfrac{df}{dt} = f_t + f_z\dfrac{dz}{dt}+f_y\dfrac{dy}{dt}+f_x\dfrac{dx}{dt}$$

$$\dfrac{df}{dt} = f_t + f_z(\dfrac{dz}{dt})+f_y(y_t + y_z\dfrac{dz}{dt})+f_x(x_t+x_y\dfrac{dy}{dt})+x_z\dfrac{dz}{dt}$$

$$\dfrac{df}{dt} = f_t + f_z(\dfrac{dz}{dt})+f_y(y_t + y_z\dfrac{dz}{dt})+f_x(x_t+x_y(y_t + y_z\dfrac{dz}{dt}))+x_z\dfrac{dz}{dt}$$

Doing the implicit partial derivatives, and doing the single-variable derivative of z, and then substituting the equations for x,y and z into the stuff that falls out, will yield the correct total (single-variable) derivative of f.

However, for some relationships of ##x,y,z,t##, the total-derivative scheme becomes complex. Examples are:

1) When there are non-linear relationships between the variables

2) When there are more than one independent variable

An example could be that the variables x,y and z are related by the following relationship, assuming that the function ##f## is now only a function of (x,y,z).:

$$A = xyz$$

When I do the total-derivative, I end up with an infinite regression of terms, because the total derivatives of one variable, depends on the total derivative of the others, which depend on the total derivative of the first one, and so on ad infinitum.

**So my question is:**

How do we deal with total-derivatives of functions where the variables are related by non-trivial (non-composites?) relationships?

How do we deal with total-derivatives of functions where the variables are related by non-trivial (non-composites?) relationships?