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A problem in the calculus of variations. Let a vessel traveling at constant speed c navigate on a body of water having surface velocity u = u(x,y) (1) v = v(x,y). (2) The ...

The

**direction**from an object A to another object B can be specified as a vector v=AB^-> with tail at A and head at B. However, since this vector has length equal to the ...An asymptotic

**direction**at a point p of a regular surface M in R^3 is a**direction**in which the normal curvature of M vanishes. 1. There are no asymptotic directions at an ...The directions in which the principal curvatures occur.

Let a be the angle between v and x, b the angle between v and y, and c the angle between v and z. Then the

**direction**cosines are equivalent to the (x,y,z) coordinates of a ...Each of the sets forming a

**direct**product is said to be a**direct**factor. A group G is said to be a**direct**factor of the group G^' if G^' is isomorphic to the group**direct**...A compositeness certificate is a piece of information which guarantees that a given number p is composite. Possible certificates consist of a factor of a number (which, in ...

The inverse limit of a family of R-modules is the dual notion of a

**direct**limit and is characterized by the following mapping property. For a directed set I and a family of ...The

**direct**sum of modules A and B is the module A**direct**sum B={a**direct**sum b|a in A,b in B}, (1) where all algebraic operations are defined componentwise. In particular, ...The differential forms on C^n decompose into forms of type (p,q), sometimes called (p,q)-forms. For example, on C, the exterior algebra decomposes into four types: ^ C = ^ ^0 ...

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