# Use of chain rule in showing invariance.

• mrmojorizing
In summary: If the resulting expression is the same (or a constant multiple of the other expression) then the equation is invariant.Note that the equation isn't invariant for most waves which we observe in the real world which is a good thing because otherwise we wouldn't observe the Doppler Shift.In summary, the conversation discusses the use of the chain rule to show that an equation is invariant under a Galilean transform. The example given is a wave equation for E, and the question is why the derivative is taken with respect to x and t rather than x' and t'. The conversation also mentions the use of the Galilean transformation to represent the motion of an object and the need to check for invariance in the wave equation.
mrmojorizing
Hi, I’m a bit confused.
I am familiar with the chain rule: if y=f(g(t,x),h(t,x)) then dy/dt=dy/dg*dg/dt+dy/dh*dh/dt
To show that an equation is invariant under a galiliean transform, it’s partially necessary to show that the equation takes the same form both for x and for x’=x-v(T). So if you have a wave equation for E which applies for x, and t, you want to show that the wave equation, with all of its first and second derivatives also applies for x’ and t’.

For example if you look at question 16 b in the following link, they ask to show that the wave equation is not invariant under Galilean transforms. What I don’t understand is in this question why are they taking the derivative of E with respect to x and t, rather than with respect to x’ and t’. We already know the wave equation takes the correct form for x and t. We want to show that it doesn’t take the correct form for x’ and t’, so then why start off taking the derivative with respect to x and t, and muck about using the chain rule rather than taking the derivative with respect to x’, and t’ (which is what you’re really interested in).

http://stuff.mit.edu/afs/athena/course/8/8.20/www/sols/sol1.pdf

I just don’t get why they take the derivative with respect to x and t, when that’s not what you’re really interested in, you’re interested in the form of the derivatives with respect to x’ and t’. Where does the idea to use the chain rule in this case comes from at all? All of the solutions for these types of problems seem to use the chain rule, I don’t get where the natural impetus to take the derivative in the wave equation with respect to variables you already know will work in the wave equations, rather than the ones which have undergone the Galilean transform.

It’s been a very long time since I’ve done this type of math/physics. Thanks.

The transformation is ##x' = x - vt,~~~~ t'=t##. This is easily inverted to give ##x=x'+vt',~~~~t=t'##.

You have a wave equation involving things like ##\partial_x^2 E##, so it's easier to simply find what that looks like in the new coordinates, and substitute that back into the original equation.

[Actually, that sect 16(b) is not completely explicit. For that, you'd have to go one step further and show that the transformed quantities are not a simple (common) multiple of the originals. I'm guessing the author probably considers that to be obvious.]

mrmojorizing said:
in question 16b when trying to find the second derivative of E wrt to t i don't understand how they go form -v(dE/dx')+dE/dt' to E*(d^2/d^2(t')+v^2*d^2/d^2(x')-2*v*d^2/(dx'*dt')).
You transcribed that incorrectly. The 2nd order derivative terms act on the ##E##, but you've moved ##E## to the front. (??)

The 1st order derivative can be written as the operator equation:
$$\partial_t ~=~ \partial_{t'} - v \partial_{x'} ~,$$where ##v## is a constant. So you just have to evaluate
$$\partial^2_t ~=~ \Big( \partial_{t'} - v \partial_{x'} \Big)^2 ~.$$Of course, to understand this properly, you've got to make both sides act on something.

(BTW, make an effort to learn how to use basic Latex for math on PF. Nobody wants to read that ugly ascii math. There's a faq under the site info menu that will help you get started.)

What you're doing is comparing the movement of the object you're interested in with the movement of the wave crest. Or rather what the wave looks like to your object if the object is moving. The Galilean transformation which strangerep stated can be used to represent the motion of the object. The wave equation in x and t can be translated to a wave equation in x' and t'. That wave equation in x' and t' is the one which applies from the object's perspective. (Or in coordinates which move with the object).

You find the wave equation in x' and t' by applying the chain for derivatives indicated by the Galilean transformation. If the two wave equations have the same form where x' can replace x and t' can replace t, then the wave equation is invariant.

Notice that you probably don't want the equation to be invariant for most waves such as sound waves. If that were the case then Doppler Shift would not occur. Doppler Shift can be determined directly from the transformed wave equation using the Galilean transformation.

What you're trying to check is whether:

$$\frac{\partial ^2 E}{\partial {x'}^2}-\frac{1}{c^2}\frac{\partial ^2 E}{\partial {t'}^2}=\frac{\partial ^2 E}{\partial {x}^2}-\frac{1}{c^2}\frac{\partial ^2 E}{\partial {t}^2}$$

You can do that by that by applying a coordinate transformation to either side of the equation to try to obtain the corresponding expression in terms of the independent variables on the other side of the equation. What they did in the reference you cited is the standard method of doing such a coordinate transformation.

## 1. What is the chain rule?

The chain rule is a mathematical rule that is used to find the derivative of a composite function. It states that the derivative of a composition of two functions is equal to the product of the derivatives of each individual function.

## 2. How is the chain rule used in showing invariance?

Invariance is the property of a function or equation that remains unchanged when certain transformations are applied to it. The chain rule is used in showing invariance by allowing us to analyze the effects of these transformations on the function and its derivative, and determine if they result in the same function or equation.

## 3. Can the chain rule be applied to any function?

Yes, the chain rule can be applied to any function that is composed of two or more smaller functions. It is a fundamental rule in calculus and is used extensively in various fields of science and engineering.

## 4. What are some examples of invariance that can be shown using the chain rule?

The chain rule can be used to show invariance in various mathematical and physical concepts. Some examples include rotational symmetry, translation invariance, and scale invariance. It can also be applied to show invariance in differential equations and optimization problems.

## 5. Are there any limitations to using the chain rule in showing invariance?

While the chain rule is a powerful tool in mathematics and science, it does have its limitations. It may not be applicable to certain types of functions, such as those that are not differentiable. Additionally, it may not be able to show invariance in highly complex or non-linear functions.

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