Clairauts “equality of mixed partial derivatives” theorem

[davidbenari]

I know how to prove this via limits and I'm okay with that.

What I want to understand is the interpretation of the theorem and specifically a visualisation of why what the theorem states must be the case.

My guess is that this theorem is saying that change is symmetrical. But I don't know if this is only true for second derivatives.

If you don't know this theorem by its name the theorem basically says this:

∂/∂y(∂f/∂x)=∂/∂x(∂f/∂y)

Also, I would like to know if you consider my focus on visualisation to be not worthwhile and that I should instead just trust this theorem.

I thank you in advance.

 

[UltrafastPED]

Clairaut's theorem (1743) is valid when the first partial derivatives are continuous. If you are a mathematics student then it is worth while understanding why this is so.

More information is available here:
https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives

and
http://calculus.subwiki.org/wiki/Clairaut's_theorem_on_equality_of_mixed_partials

 

[davidbenari]

I am a physics student. I have a question about this "if you are a mathematics student then it is worthwhile" stuff. Would a great physicist, say like Feynman, know why this is so? Even if your answer is speculative, what do you think?

 

[AlephZero]

If the math is a model of some physical situation, you can often argue that derivatives are continuous, etc, on physical grounds. Usually, a math model only hits the "exceptional" conditions that mathematicians like to understand completely, if it's a poor model of the physics.

IMO it is always worthwhile (not to say essential) to know something about the conditions that make your math valid. But as a physicist or engineer you don't necessarily need to know the most general set of conditions that make it valid, or be able to prove why it is valid.

Of course you can never know "too much" math, but in real life, whether you learn more math or more physics is a time management problem.

(Full disclosure: I've seen both sides of this first hand - I have a math degree, and spent most of my life working on engineering problems).

 

[CellCoree]

The Clairauts "equality of mixed partial derivatives" theorem is a fundamental result in multivariable calculus that states that the order of differentiation of a function with respect to two different variables does not matter. In other words, the mixed partial derivatives of a function are equal. This theorem is important because it allows us to simplify calculations and make predictions about the behavior of a function in multiple dimensions.

To understand the interpretation of this theorem, it is helpful to visualize the concept of change in a function. Imagine a surface representing a function, with one variable along the x-axis and another variable along the y-axis. The mixed partial derivatives represent the rate of change of the function as we move along the surface in different directions. The Clairauts theorem tells us that the rate of change in one direction is equal to the rate of change in the other direction, regardless of the order in which we take the derivatives.

This theorem is not limited to just second derivatives, but applies to all orders of mixed partial derivatives. It essentially states that the function is "symmetrical" in terms of its rate of change in different directions.

As for the focus on visualization, it can definitely be helpful in understanding mathematical concepts. However, it is also important to trust and understand the proof of the theorem, as it provides a rigorous justification for its validity. So it is beneficial to have a balance of both visualization and understanding of the mathematical reasoning behind the theorem.

In conclusion, the Clairauts theorem is a powerful tool in multivariable calculus that allows us to simplify calculations and make predictions about functions in multiple dimensions. Its interpretation as a statement of symmetry in the rate of change of a function is a helpful visualization, but it is also important to understand the proof and trust in the validity of the theorem.