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A bundle map is a map between bundles along with a compatible map between the base manifolds. Suppose p:X->M and q:Y->N are two bundles, then F:X->Y is a bundle map if there ...

A vector field is a map f:R^n|->R^n that assigns each x a vector f(x). Several vector fields are illustrated above. A vector field is uniquely specified by giving its ...

Trigonometry

The derivative of a function represents an infinitesimal change in the function with respect to one of its variables. The "simple" derivative of a function f with respect to ...

An (ordinary) torus is a surface having genus one, and therefore possessing a single "hole" (left figure). The single-holed "ring" torus is known in older literature as an ...

An alternating permutation is an arrangement of the elements c_1, ..., c_n such that no element c_i has a magnitude between c_(i-1) and c_(i+1) is called an alternating (or ...

If an n-sided Poncelet transverse constructed for two given conic sections is closed for one point of origin, it is closed for any position of the point of origin. ...

The hyperbolic cotangent is defined as cothz=(e^z+e^(-z))/(e^z-e^(-z))=(e^(2z)+1)/(e^(2z)-1). (1) The notation cthz is sometimes also used (Gradshteyn and Ryzhik 2000, p. ...

A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable. This definition is ...

A diffeomorphism is a map between manifolds which is differentiable and has a differentiable inverse.