Proof that if x-partial derivative and y-partial derivative are equal, the function depends only on x+y [Wanmin Math Memo]

Today, we will take up the following problem. It is a proof that for a function of two variables, if its x-partial derivative and y-partial derivative are equal, then the function depends only on x+y.

Introduction

By the way, this problem is covered in the following video.

▼ [Mathematics Exercises for Transfer Exams Chapter 6: Partial Differentiation] Example 6-7. Chain Rule (4): Proof Problems "Thorough Research on Transfer Mathematics"

In this video, by treating (1) as a guide,

$$
u=x+y, \ v = x- y
$$

the proof is carried out using the variable transformation above.

However, is it impossible to prove it using any other variable transformation?

Below, I will provide a general proof.

General Proof

And so, the proof is complete.

Summary

In fact, for any variable transformation, as long as we set u=x+y, z_v will be 0.

During the proof, we assume the existence of an inverse matrix, but for this to exist, it is sufficient that the Jacobian is not 0, which is a natural condition for a "well-behaved variable transformation."