n. 1. degree of arc. a measure of angle equal to one 360th part of the angle traced out by one full revolution of a line segment around one of its endpoints, written °. One degree is divided into 60 minutes, or 3600 seconds. Compare radian. 2. the highest power or sum of powers in any term of a given polynomial or algebraic equation, or the sum of powers in any one term. For example, x⁴ + 3x²- x and xy²z are both of the fourth degree. See also quadratic, cubic, quartic, etc. 3. the greatest power of the derivative of highest order in a differential equation. For example, D₃² + D₂³+ D₁⁴ = 0, where D_i is the i^th derivative, is a second degree differential equation. Compare order. 4. (of a representation of a group) the degree of the general linear group over a field into which the representation is a homomorphism from the given group. 5. (of a vertex in graph) the number of coincident edges at the given vertex. In a network or digraph, entering arcs (the in-degree) and exiting arcs (the out-degree) are counted separately. 6. (Topology) another word for genus. 7. topological degree. (for a continuously differentiable function, f, on Euclidean space) the excess of the number of points of a given region, G, in f^-1(a) at which the Jacobian is positive over those at which it is negative; this is referred to as the degree of f at a in region G, and written D[a, G, f]. More generally, this extends to a number (known as Brouwer's form of the degree for a continuous function f defined on G) that is a topological invariant that, when non-zero, ensures that f(x) = a has solution in G. 8. (of an extension field with respect to a base field) the dimension of the extension viewed as a vector space over the base field. 9. (of membership of a set) see fuzzy set theory.