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sgsawant

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_{xy}

2. f

_{yx}

Are the above 2 derivatives equal, in general. Please explain if you know the answer.

Regards,

-sgsawant

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- Thread starter sgsawant
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- #1

sgsawant

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2. f

Are the above 2 derivatives equal, in general. Please explain if you know the answer.

Regards,

-sgsawant

- #2

phyzguy

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HallsofIvy

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Landau

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Even better:

Let U in R^2 be open, [itex]f:U\to\matbb{R}[/itex] partial differentiable w.r.t. both variables, and [itex]D_1f[/itex] partial differentiable w.r.t the second variable. Suppose further that (x,y) is in V, and [itex]D_2D_1f[/itex] is continuous at (x,y). Then [itex]D_2f[/itex] is partial differentiable w.r.t the first variable at (x,y), and

[tex]D_1D_2f(x,y)=D_2D_1f(x,y)[/tex].

i.e. we only need [itex]D_2D_1f[/itex] to exist and be continuous at some point in the interioir, this already implies that [itex]D_1D_2f[/itex] exists at that point and the two are equal.

Let U in R^2 be open, [itex]f:U\to\matbb{R}[/itex] partial differentiable w.r.t. both variables, and [itex]D_1f[/itex] partial differentiable w.r.t the second variable. Suppose further that (x,y) is in V, and [itex]D_2D_1f[/itex] is continuous at (x,y). Then [itex]D_2f[/itex] is partial differentiable w.r.t the first variable at (x,y), and

[tex]D_1D_2f(x,y)=D_2D_1f(x,y)[/tex].

i.e. we only need [itex]D_2D_1f[/itex] to exist and be continuous at some point in the interioir, this already implies that [itex]D_1D_2f[/itex] exists at that point and the two are equal.

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