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 April 5th, 2009, 02:11 PM #1 Newbie   Joined: Apr 2009 Posts: 2 Thanks: 0 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
 April 5th, 2009, 03:15 PM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Quick question In general, we can't say anything about any relationship. Note that, dimensionally speaking, where $\left[D\right]$ defines the dimensions of a term $D,$ $\left[\frac{\partial^2f}{\partial x\partial y}\right]=\frac{\left[f\right]}{\left[x\right]\left[y\right]},$ whereas $\left[\frac{\partial f}{\partial x}\frac{\partial f}{\partial y}\right]=\frac{\left[f\right]^2}{\left[x\right]\left[y\right]},$ so we wouldn't expect any meaningful identity between the two. One relationship that does involve the two different derivatives is $\frac{\partial}{\partial x}\frac{\partial}{\partial y}\left(\frac12f^2\right)=\frac{\partial}{\partial x}\left(f\frac{\partial f}{\partial y}\right)=\frac{\partial f}{\partial x}\frac{\partial f}{\partial y}+f\frac{\partial^2f}{\partial x\partial y}.$
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 $\left[D\right]$ defines the dimensions of a term $D,$ $\left[\frac{\partial^2f}{\partial x\partial y}\right]=\frac{\left[f\right]}{\left[x\right]\left[y\right]},$ whereas $\left[\frac{\partial f}{\partial x}\frac{\partial f}{\partial y}\right]=\frac{\left[f\right]^2}{\left[x\right]\left[y\right]},$ so we wouldn't expect any meaningful identity between the two. One relationship that does involve the two different derivatives is $\frac{\partial}{\partial x}\frac{\partial}{\partial y}\left(\frac12f^2\right)=\frac{\partial}{\partial x}\left(f\frac{\partial f}{\partial y}\right)=\frac{\partial f}{\partial x}\frac{\partial f}{\partial y}+f\frac{\partial^2f}{\partial x\partial y}.$

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)

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 $\left[D\right]$ defines the dimensions of a term $D,$ $\left[\frac{\partial^2f}{\partial x\partial y}\right]=\frac{\left[f\right]}{\left[x\right]\left[y\right]},$ whereas $\left[\frac{\partial f}{\partial x}\frac{\partial f}{\partial y}\right]=\frac{\left[f\right]^2}{\left[x\right]\left[y\right]},$ so we wouldn't expect any meaningful identity between the two. One relationship that does involve the two different derivatives is $\frac{\partial}{\partial x}\frac{\partial}{\partial y}\left(\frac12f^2\right)=\frac{\partial}{\partial x}\left(f\frac{\partial f}{\partial y}\right)=\frac{\partial f}{\partial x}\frac{\partial f}{\partial y}+f\frac{\partial^2f}{\partial x\partial y}.$

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)

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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