Local and integral physical quantities

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Hello everyone,
I'd like to know if my understanding of local and integral quantities is clear.

An integral quantity refers to the entire physical system, it is not defined point by point.
A local one is defined point by point, for example ρ(x,y,z).

Can I consider the charge dq as a local quantity? It is the charge of an infinitesimal element of something.
I can integrate it and find the integral quantity q.

In some quizzes my prof asks to write three or four local and integral quantities.
Integral quantities could be mass, electric potential, resistance, charge.
Local quantities could be resistivity, electric field, density or the density of current.

I'm not sure if I can consider local quantities the differential quantities like dL, dm, dq, di, dV. The correspondent integral ones are lenght, mass, charge, intensity of current, potential.

Thank you for your answers
 

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Simon Bridge
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What is the context?
 
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DEvens
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Usually local and integral are not technical terms. They mean just what the English words mean. The value at a location is the local value. The total or integrated value is the integral value. So things like mass density and total mass, charge density and total charge, and so on, are just related by the appropriate integral.

Sometimes the relationship between the local value and the bulk or integrated value can be complicated. Viscosity, for example, is a slippery concept. (Sorry.) The viscosity of a sample 1 mm thick may, or may not, be simply related to the value you would measure from a sample 10 mm thick, or 100 mm thick. Young's modulus is sometimes complicated in this fashion. A single crystal of a material may have one resistance. But a jumbled pile of single crystals all crushed together, as some materials are, may have a very different resistance.There are some other properties that may be complicated this way. In these cases there is some physics going on that means these values do not behave linearly with distance.

There are some properties that are not local. For example, topological properties are not local. For example, consider a long strip of paper that has been joined into a loop. If you join it one way you get a simple tube. If you give it a 180 degree twist before you join it you get a Mobius strip. Two 180 degree twists you get another shape (for which I'm not sure if there is an agreed upon name). The zero-twist and two-twist shapes have each got two well defined sides. But the Mobius strip has only one side. You can get to the "other side" by taking a path that never leaves the surface of the paper. But you can't tell this by looking at only a small patch of the shape. So the number of disconnected sides a strip of this kind has is a non-local property. Locally it looks like a 2-D surface with two sides. Globally it may be different.
 

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