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An affine tensor is a tensor that corresponds to certain allowable linear coordinate transformations, T:x^_^i=a^i_jx^j, where the determinant of a^i_j is nonzero. This ...

The functions describing the horizontal and vertical positions of a point on a circle as a function of angle (cosine and sine) and those functions derived from them: cotx = ...

A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which ...

The Machin-like formula 1/4pi=cot^(-1)(2)+cot^(-1)(3). The other 2-term Machin-like formulas are Hermann's formula, hutton's formula, and Machin's formula.

The exterior derivative of a function f is the one-form df=sum_(i)(partialf)/(partialx_i)dx_i (1) written in a coordinate chart (x_1,...,x_n). Thinking of a function as a ...

Let f:R×R->R be a one-parameter family of C^2 map satisfying f(0,0)=0 [(partialf)/(partialx)]_(mu=0,x=0)=0 [(partial^2f)/(partialx^2)]_(mu=0,x=0)>0 ...

Gauss's theorema egregium states that the Gaussian curvature of a surface embedded in three-space may be understood intrinsically to that surface. "Residents" of the surface ...

Johnson's theorem states that if three equal circles mutually intersect one another in a single point, then the circle passing through their other three pairwise points of ...

For every ring containing p spheres, there exists a ring of q spheres, each touching each of the p spheres, where 1/p+1/q=1/2, (1) which can also be written (p-2)(q-2)=4. (2) ...

Let a triangle have sides of length a, b, and c and let the angles opposite these sides be denoted A, B, and C. The law of tangents states ...