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2025, Vol. 10, Issue 1, Part A

A study on symmetry of second derivatives with symmetric functions


Author(s): Ravi Kumar Verma

Abstract:
The link between the two results is their equivalence, which is established in Theorem. For the sake of simplicity we confine our discussion to functions defined on open sets of the plane and to second order mixed partial derivatives. The interested reader can easily extend the obtained result to more general situations. All functions of two variables are assumed to be continuous ("jointly continuous" is the terminology preferred by some authors). Marsden and Hoffman use the Mean Value Theorem to give a plausible proof of the equality xy = yx while emphasizing its nonintuitive nature. Kaplan derives the same equality from Fubini's Theorem in the case when f, fx, fy and fxy are continuous. A first order differential equation: M(x,y)+N(x,y)(dy/dx)=0, often written as M(x,y)dx + N(x,y)dy = 0, is said to be exact, over a domain D in R^2, if there exists a continuously differentiable function f(x,y), such that the functions M and N are the partial derivative functions of f(x,y), with respect to x and y respectively, i.e. M(x,y)=(f_x)(x,y) and N(x,y)=(f_y)(x,y) for every (x,y) in D.
The most important characterization of exact equations requires that the above function f(x,y) have identically equal mixed second order derivatives f_ x,y=f_ y,x. A standard sufficient condition for the above is continuity of second order partial derivatives. This condition states that the above condition is exact if and only if the first order partials M_ y(x,y)=N_ x(x,y) for every (x,y) in D.


DOI: 10.22271/maths.2025.v10.i1a.1950

Pages: 45-48 | Views: 72 | Downloads: 10

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International Journal of Statistics and Applied Mathematics
How to cite this article:
Ravi Kumar Verma. A study on symmetry of second derivatives with symmetric functions. Int J Stat Appl Math 2025;10(1):45-48. DOI: 10.22271/maths.2025.v10.i1a.1950

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