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Point density

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I'm starting to read about radiometry and see I need to review some calculus stuff.  I'm trying to understand density at a small length (keeping it 1D for simplicity). 

 

I've seen density at a point defined by:

 

lim_{dL->0} deltaM/deltaL

 

So if I want to know the density at a point x, I might form a small segment about x:

 

deltaL = (x+h) - x = h

 

I understand conceptually that deltaM needs to be the average mass in this small segment to get the local density (then in the limit it becomes a point density).  But what is the formula for deltaM?  I feel like it should be something like:

 

deltaM = m(x+h) - m(x)

 

but what is m(x)?  At first I started by assuming m(x) is a mass function (it gives me the mass at point x).  But that doesn't make sense.  The difference between two point masses doesn't give me the average mass in the segment. 

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I understand conceptually that deltaM needs to be the average mass in this small segment

Not the average mass - the difference in mass between m(x) and m(x+delta).

 

 

 


The difference between two point masses doesn't give me the average mass in the segment.

Correct. It gives you the difference in mass as mentioned above. Remember that you then divide that by a difference in distance (deltaL) in your limit function to get mass-per-unit-length, which is what you're trying to determine.

 

A general description of what you're doing is determing the mass in the delta-length of interest, the mass within the the delta-length. In the illustration below, m(x) is the mass at position x. When you subtract the m(x) from m(x+delta) you get the mass (the red area) under the curve. EDIT: I've got the axes messed up a bit, but hopefully you get the idea.

density01.png

Edited by Buckeye
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I am afraid Buckeye's post doesn't make things clear. The red area is the integral of m between x and x+delta, not just m(x+delta)-m(x).


I've seen density at a point defined by:
 
lim_{dL->0} deltaM/deltaL

This is a horrible way to write it: "dL->0" means nothing.

If you have a function M such that M(x) tells you how much mass there is to the left of x, then the mass density is the derivative of M.

lim_{h->0) (M(x+h) - M(x)) / h
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This is a horrible way to write it: "dL->0" means nothing.

 

How is dL->0 different from h->0.  My dL is the same as your h.

 


If you have a function M such that M(x) tells you how much mass there is to the left of x, then the mass density is the derivative of M.

lim_{h->0) (M(x+h) - M(x)) / h

 

This works because then M(x+h) - M(x) tells you the mass in the "cell" over h. 

 

What is the generalization to 2D area densities--M(x,y) where there is not a "left" or "right"?

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This is a horrible way to write it: "dL->0" means nothing.

 
How is dL->0 different from h->0.  My dL is the same as your h.


The difference is that the notation dL is standard in math to represent the exterior derviative of L, which is something hard to explain, but not a variable used to define a limit.

If you have a function M such that M(x) tells you how much mass there is to the left of x, then the mass density is the derivative of M.

lim_{h->0) (M(x+h) - M(x)) / h

 
This works because then M(x+h) - M(x) tells you the mass in the "cell" over h. 
 
What is the generalization to 2D area densities--M(x,y) where there is not a "left" or "right"?


M(x,y) can be the mass that is to the left of x and below y. You can then define the mass density as

lim_{h->0} lim_{l->0) (M(x+h,y+l) - M(x+h,y) - M(x,y+l) + M(x,y)) / (h * l)
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