![]() ![]() In any coordinate system, I can define an interval whose unit length I can define, right eg. My argument for the comparison of volume elements across different coordinates systems: I would highly appreciate people from a physics background to answer this question in an intuitive, "Feynman lectures" way, for lack of better words. This is the infinitesimal volume in Cartesian coordinates.I.e., an infinitesimal volume should have all it's edges as infinitesimal lengths, right? Is any other infinitesimal volume theoretically correct? (I have trouble accepting cuboid-shaped "infinitesimals" as well.) If then I multiply this length with the same but in two perpendicular directions, I get a cube. And if we said u f ( x) we also needed to find d u that corresponed. When we made a u substitution in plane old single variable calcululs we were effectively making a change of the coordinate system. Infinitesimals don't exist in the traditional real number system, but they do in a variety of other systems, including unreal and hyperreal numbers, which are real numbers augmented with a system of infinitesimal quantities, and infinite quantities, which are the reciprocals of the infinitesimals. When we convert to polar, a small change in wich we will call d corresponds to an arc length of r d. An infinitesimal is by definition a length that is really, really small. The infinitesimal definition in mathematics is The quantities that are closer to zero than any standard real number but are not zero are Infinitesimals.Is it OK to address the infinitesimal volumes as smaller versions of finite shapes? If it is fine, what is wrong in this Gedanken?: This involves a phrasing of what infinitesimals are, how an infinitesimal volume arises, and what happens when such volumes are compared from two different coordinate systems. But please note that the question title has not been changed at all! It stands.)Īfter the many discussions, now the questions stand at comparing infinitesimal volumes.Ī holistic answer that addresses this will be appreciated. (Sorry if people thought that it meant: "Is it possible and is it done in daily life to use anything other than the Cartesian volume element?" : I know the answer to this is of course yes and I know it's usefulness. The question clearer: Is the infinitesimal cube the absolute smallest infinitesimal volume? ![]()
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