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Central field in euclidean spaces

A C -vector field over R \ {0} is called a central field if V (T (p)) = T (V (p)) (T ∈ O (n, R)) {\displaystyle V(T(p))=T(V(p))\qquad (T\in \mathrm {O} (n,\mathbb {R}))} where O(n, R) is the orthogonal group. We say central fields are invariant under orthogonal transformations around 0.

The point 0 is called the center of the field.

Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.