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3

People have constructed three-dimensional convex objects with only one stable point. These are called gömböcs (if you want to be pedantic, the Hungarian plural is gömböcök). It is straightforward to use these to obtain polyhedra with only one stable side. Polygons and polyhedra must have at least one stable side. Suppose we have a polygonal or polyhedral ...


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For an open orientable surface there are two possible, equivalent normals: $\vec n$ and $-\vec n$. The usual convention is that you choose a direction in which the perimeter of the surface is traversed and define the positive direction as the direction given by the right hand rule, as shown in the following picture. This can also be done for non-simply ...


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By convention, for a flat lamina or a plane surface, the area vector is a vector whose magnitude is the area of the surface and whose direction points in a direction perpendicular to the surface. If you have a curved surface, then you have to consider elemental areas, i.e: small patches of area denoted by $dA$ whose direction is perpendicular to the small ...


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They are linked by the "law of the right hand": The preferred direction of dℓ⃗ dℓ→ along the loop is that from the palm to fingertips of your right hand when it surrounds the loop. Then, the associated preferred direction of dA⃗ dA→ is indicated by the thumb.


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There's a plain distinction between a reference system (to use a more contemporary designation) and a coordinate system: a coordinate system is a reference system together with an (one-to-one) assignment of a coordinate value ($n$-tuple) to each element. Expressed more formally, a reference system to begin with is constituted by a set of distinguishable, ...


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The "system of coordinates" is just a fancy way to describe particular "coordinates", which you said to now what it means, with the focus on the "choices" which were made to choose some way of defining and measuring coordinates and not others. "System of reference" was just a particular phrase used by Einstein once, it is not routinely used, and what he ...



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