Search results
Results From The WOW.Com Content Network
Free product. In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “ universal ” group having these properties, in the sense that any two ...
Generalizing the statement above, for a family of path connected spaces , the fundamental group () is the free product of the fundamental groups of the . [10] This fact is a special case of the Seifert–van Kampen theorem, which allows to compute, more generally, fundamental groups of spaces that are glued together from other spaces.
Seifert–Van Kampen theorem. In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen ), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space in terms of the fundamental groups of two open, path-connected subspaces ...
Path (topology) The points traced by a path from to in However, different paths can trace the same set of points. In mathematics, a path in a topological space is a continuous function from a closed interval into. Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there ...
In algebraic topology, the path space fibration over a pointed space [ 1] is a fibration of the form [ 2] where. is the based path space of the pointed space ; that is, equipped with the compact-open topology. is the fiber of over the base point of ; thus it is the loop space of . The free path space of X, that is, , consists of all maps from I ...
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. [ 1] A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the ...
The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on ...
Free abelian group. In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer ...