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April 5th, 2009, 02:11 PM   #1
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Quick question

I am working on something for my dissertation and I need to know if I can establish any relationship/bound on the difference between a cross derivative and the product of partial derivatives, that is,

d2f/dxdy (=,<,>?) df/dx*df/dy

(Any reference?)

This must be super-obvious, but I thought that the asking was going to be the mos efficient way of finding out.

Thanks,
DD
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April 5th, 2009, 03:15 PM   #2
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Re: Quick question

In general, we can't say anything about any relationship. Note that, dimensionally speaking, where defines the dimensions of a term

whereas

so we wouldn't expect any meaningful identity between the two.

One relationship that does involve the two different derivatives is

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April 5th, 2009, 04:40 PM   #3
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Re: Quick question

Quote:
Originally Posted by mattpi
In general, we can't say anything about any relationship. Note that, dimensionally speaking, where defines the dimensions of a term

whereas

so we wouldn't expect any meaningful identity between the two.

One relationship that does involve the two different derivatives is


This is great thanks very much. If you don't mind I am going to ask something else related to what you replied. The function f I am working with is between [0,1], can't that help me saying that d2f/dxdy>df/dx*df/dy? (from your answer this seems to be a correct statement, but I am not sure)

Thanks once again for your reply
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April 8th, 2009, 08:44 AM   #4
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Re: Quick question

Quote:
Originally Posted by dabdias
Quote:
Originally Posted by mattpi
In general, we can't say anything about any relationship. Note that, dimensionally speaking, where defines the dimensions of a term

whereas

so we wouldn't expect any meaningful identity between the two.

One relationship that does involve the two different derivatives is


This is great thanks very much. If you don't mind I am going to ask something else related to what you replied. The function f I am working with is between [0,1], can't that help me saying that d2f/dxdy>df/dx*df/dy? (from your answer this seems to be a correct statement, but I am not sure)

Thanks once again for your reply
I find this entire place kind'a confusing, but in general we can't expect alot. That is, a function of two variables need an explaination of how the respective set interact, here assume the real with it usual topology. All we know is that it is a function and that the produce under consideration is the cross the product. It is therefore must be differentiable with respect to a direct product of R in the sense descibed. The richness of tensor and the likes come, usually, as one "zoom in" consider such a large collection of such function. Let just consider the simple case of one variable, the class of pointwise produce h*g is member hence

(h*g)' = h'g +g'h for a strong "central" domain in R.

lim (h(x +h)g(x + h) - h(x)g(x))/h must be consider when x is not a point due to the nature of the topology need in the usage of limits. h is a variable representing "a moving number." Kind of like a vibrational slope, "its vibrates at a singularity."
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