Mechanics of materials dx,dz,dz

[Cyrus]

Im taking mechanics of materials. One of the things they talk about is cutting out a small elemental cube of a rigid body, that has sides dx,dz,dz. Is it always true that dy,dx, and dz have the same infinitesimal size? I thought that they would not necessarily be the same size, which could give you a rectangle. The reason I thought this is say you have say a rectangular box, and cut it with a grid pattern, and you make the grid finer and finer. Then if its longer in the x direction than the y direction, a rectangular box, and I make my grid all squares based on the smallest dimenson, the y direction, then I can shrink all the squares more and more. It is clear that as my grid shrinks, I will approach dy much faster than I approach dx. I would expect to get to dy first, as y is the smaller direction, and dx much later, if its x>>y, since I cut it into cubes and made those cubes finer and finer.

 

[cepheid]

I'm not sure I follow. Why can't you cut an elemental cube out of any volume you want? Why does it matter if it is a rectangular box? We said we were looking at an elemental cube. Therefore, it is a cube.

Also, can't an infinite number of such infinitesimal cubes make up any volume of any shape?

Sorry to respond to questions with questions, but I'm not 100% positive.

 

[Cyrus]

Its cool. It matters if its an rectangular box because Mohrs Circle works for an elemental cube, not a rectangle. I am asking if dy, dx and dz are always equal in value. I have never read anywhere that said they were, and usually engineering texts are very loose in how they use their math. I know that dx, dy and dz are independent of each other, so I thought they might also be different in value from one another, but I was not quite sure.

 

[dx]

dx, dy or dz or any other infinitesimals are not finite quantities. You cannot assign a definite value to them and you cannot compare their sizes. For physics and engineering, you can think of them as 'sufficiently small' quantities (so that you get the accuracy you desire, or you can pass to the limit in an ideal situation). This is of course, not mathematically rigorous. Take the pythagorean theorem for example. On a curved 2 dimensional surface, $$ds^2 = dx^2 + dy^2$$ describes the geometry of the surface at a 'sufficiently small' area. Hm.. actually, I am kinda confused.. the above equation seems to imply that $$ds^2$$ is somehow larger than the other two.. but that would be meaning less, ds is an infinitesimal length, just like the other two..help..

 

[D H]

Look into nonstandard analysis, which attempts to make the loosey-goosey handwaving made by physicists rigorous. Infinitesimals are comparable/scalable/etc. Without these features, finding the length of a curve is downright difficult.