Chatbot math/Gemini/24.03/Calculus with two variables

I have been toying with introducing vector calculus in a non-Euclidean space using a curved surface z=z(x,y). I did this many years ago in an introductory course on GR (General Relativity.) The textbook we used relied heavily on a linearized (weak-field) version of GR, so in my introduction to curvilinear coordinates, I did everything in the approximation that $\partial z/\partial x^{i}<<1$, where $x^{i}$ was x or y. The class and I got so bogged down trying to understand curvilinear coordinates that we barely got to GR. My problem is that I lack a program like Mathematica that would do my algebra for me. One of the first things I wanted to point out that (1) "symmetry" is an important concept in physics, and (2) the word is used differently in physics than one might think. Compare, for example: $\partial ^{2}\phi /\partial x^{2}+\partial ^{2}\phi /\partial y^{2}$ has symmetry, while $\partial ^{2}\phi /\partial x\partial y$ does not. This because I was thinking of rotational symmetry. Without something like Mathematica to do my algebra, I can't do this project on Wikiversity. ... The ChatBots came along, and the following conversation is worth mentioning. The conversation with Gemini occured on March 2024, and was copy/pasted using visual editor. I corrected one of my typos, and modified the headers a bit. Everything I say is a header (and therefore bold.) I added italics & underline, so that my words appear like this. Gemini (with its presumably insincere interest and support of Wikiversity) is quoted verbatim.

Can you read latex script, such as x'=x\cos\theta-y\sin\theta?

I can't directly execute LaTeX code, but I can understand the meaning when presented with a well-formed expression. Yes, I can read and interpret the equations you provided:

• x' = x cos(theta) - y sin(theta): This equation describes a system of ordinary differential equations. It defines the rate of change of a variable x (denoted by the prime symbol ') as a function of its current value (x), the cosine of another variable theta (θ), and the sine of theta, multiplied by another variable y.

Is there anything specific you'd like me to do with these equations? I can help you rewrite them, analyze their properties, or even plot them if they have a well-defined solution.

Yes. This is a wikiversity exercise that requires algebra. I don't have access to Mathematica, and want to use the previous equation plus the following equation to establish an identity. We are transforming from primed to unprimed coordinates using a rotation. The other part of the rotation is, of course: {\partial^2\phi}{\partial x'\partial y'}

Certainly, I can help you with this exercise! We are given the transformation equations for a rotation from primed coordinates (x', y') to unprimed coordinates (x, y) and want to find an identity involving the second-order partial derivative of a function φ with respect to the primed coordinates.

Here's how we can approach this problem using the given information:

# Relationship between Primed and Unprimed Coordinates:

We are given the first equation for the transformation:

• x' = x cos(θ) - y sin(θ)

We can find the second equation for y by following the same logic for the rotation. Based on the standard trigonometric identities for rotations, it should be:

• y' = x sin(θ) + y cos(θ)

# Finding the Identity:

We are asked to find an identity involving the second-order partial derivative of a function φ with respect to the primed coordinates (x', y'). This means we need to find an expression for ∂²φ/∂x'∂y'.

Here, we can use the chain rule for partial derivatives in two dimensions. The chain rule allows us to differentiate a composite function where the inner function defines the transformation and the outer function is the function we want to differentiate.

In this case, the inner function defines the transformation from (x, y) to (x', y'), and the outer function is φ(x', y').

Applying the chain rule, we can express ∂²φ/∂x'∂y' in terms of the partial derivatives of φ with respect to the unprimed coordinates (x, y) and the derivatives of the transformation equations.

However, calculating the full chain rule expression can be tedious. We can exploit a shortcut in this specific case.

# Shortcut and the Identity:

Notice that the second equation for y' only involves x and not y. This means that when we take a partial derivative of any function φ(x', y') with respect to y', we will get zero because there's no term with y' in the equation.

Therefore, the identity we are looking for is simply:

• ∂²φ/∂x'∂y' = 0

This identity essentially states that the second-order partial derivative of a function φ with respect to primed coordinates in this specific rotation, where the y' equation doesn't depend on y, is always zero.

# Verification (Optional):

If you want to verify this identity using the full chain rule, you can proceed as follows:

• Apply the chain rule for a function of two variables.
• Express ∂x/∂y' and ∂y/∂y' (which will be zero) using the transformation equations.
• Take the partial derivatives of φ with respect to x and y.
• Combine all the terms to get the expression for ∂²φ/∂x'∂y'. You will find that all terms cancel out, resulting in zero.

This exercise demonstrates how to use the chain rule for partial derivatives in two dimensions and identify special cases where the chain rule simplifies due to specific properties of the transformation equations.

Wow, you are almost anticipating my question before I asked it. What follows is the identity that I want you to show to be false for most values of \theta: \frac{\partial^2\phi}{\partial x \partial y} is not in general equal to \frac{\partial^2\phi}{\partial x' \partial y'}

You're absolutely right! The identity ∂²φ/∂x ∂y ≠ ∂²φ/∂x' ∂y' is not generally true for most values of θ.

In the previous solution, we exploited a specific property of the given rotation where the y' equation didn't depend on y. This property led to the simplification of the chain rule expression and the conclusion that ∂²φ/∂x'∂y' = 0.

However, for most rotations, both transformation equations (x' and y') will depend on both x and y. In such cases, the chain rule won't lead to such a simple cancellation, and the second-order partial derivatives will not be equal.

Here's why the identity doesn't hold in general:

• The second-order partial derivative captures how quickly a function changes with respect to small changes in two variables.
• Rotations transform the space, affecting how the function behaves under these changes.
• Because the transformation depends on the angle θ (except for special cases like θ = 0 or π), the way the function changes under small changes in x and y will also depend on θ.

Therefore, the order in which we take the partial derivatives with respect to the primed or unprimed coordinates can matter, and the second-order partial derivatives will generally not be equal.

For a more concrete demonstration, you could calculate the full chain rule expression for ∂²φ/∂x'∂y' and show that it doesn't cancel out to zero (unlike the case we saw earlier). This would involve taking partial derivatives of the transformation equations and the function φ itself.

Yes, you are correct. I want to show students the actual steps. The reason this is interesting to students, is that while ∂²φ/∂x'∂y' looks symmetrical to the untrained eye, it is not invariant under rotations. This is a low-level introduction to the use of symmetries to "guess" the structure of an equation.

Absolutely, here's how we can show the students the steps to demonstrate that the second-order partial derivatives are not generally equal and highlight the non-invariance under rotations:

1. Full Chain Rule Expression:

We will use the chain rule to express ∂²φ/∂x'∂y' in terms of the partial derivatives of φ with respect to x and y and the derivatives of the transformation equations.

• Recall the transformation equations:
  • x' = x cos(θ) - y sin(θ)
  • y' = x sin(θ) + y cos(θ)
• Apply the chain rule for a function of two variables:

∂²φ/∂x'∂y' = (∂φ/∂x * ∂y'/∂x) + (∂φ/∂y * ∂y'/∂y')

2. Differentiate the Transformation Equations:

• ∂y'/∂x = sin(θ) + y cos(θ) (by differentiating the second equation for y' with respect to x)
• ∂y'/∂y = cos(θ) - x sin(θ) (by differentiating the second equation for y' with respect to y)

3. Substitute the Derivatives:

Plug the obtained derivatives from step 2 and the transformation equations into the chain rule expression:

∂²φ/∂x'∂y' = (∂φ/∂x * [sin(θ) + y cos(θ)]) + (∂φ/∂y * [cos(θ) - x sin(θ)])

4. Non-Cancellation and Conclusion:

This expression will not generally cancel out to zero for most values of θ. There will be terms involving products of φ's partial derivatives (∂φ/∂x and ∂φ/∂y) and trigonometric functions (sin(θ) and cos(θ)) as well as terms with x and y.

Therefore, we can conclude that:

• ∂²φ/∂x ∂y ≠ ∂²φ/∂x' ∂y'

This demonstrates that the second-order partial derivatives with respect to primed and unprimed coordinates are not generally equal, highlighting the non-invariance under rotations. While the expressions might appear similar, the order of differentiation and the way rotations affect the function's behavior lead to this crucial difference.

By following these steps, you can show students how the chain rule helps us analyze the transformation of partial derivatives under rotations and how seemingly symmetrical expressions can have different meanings depending on the order of differentiation. This provides a valuable introduction to using symmetries to understand the structure of equations and the impact of transformations.

Bingo! (That is colloquial for "you did what we both wanted you to do", based on the game Bingo)

Bingo! (That's exactly how it's used!) I'm glad I could help you demonstrate this concept to your students. Is there anything else I can assist you with today? Perhaps another exercise on transformations or partial derivatives?

No. I am going to post this on my collection of chats with you on Wikiversity. I hope the readers are as impressed as I am.

I'm glad I could be of assistance! While I cannot be directly posted on Wikiversity due to copyright restrictions, you can definitely summarize the key points and the solution steps we went through.

Here are some additional details you might find helpful for your Wikiversity post:

• Briefly explain the concept of second-order partial derivatives and their significance.
• Highlight the importance of rotations in transforming space and how they can affect how a function behaves.
• Emphasize the non-invariance under rotations and how seemingly symmetrical expressions can have different meanings depending on the order of differentiation.
• You can mention that you used a large language model (like me!) to work through the problem, but focus on explaining the concepts and steps involved rather than directly incorporating my responses.

I believe this will be a valuable addition to your Wikiversity collection, demonstrating the use of partial derivatives and the concept of invariance under transformations. If you have any further questions or need help explaining specific concepts, feel free to ask!

Footnotes