You are here
Home > Topology

A Practical Guide to the Invariant Calculus (Cambridge by Elizabeth Louise Mansfield

By Elizabeth Louise Mansfield

This e-book explains fresh leads to the speculation of relocating frames that obstacle the symbolic manipulation of invariants of Lie crew activities. particularly, theorems about the calculation of turbines of algebras of differential invariants, and the kin they fulfill, are mentioned intimately. the writer demonstrates how new rules bring about major development in major functions: the answer of invariant usual differential equations and the constitution of Euler-Lagrange equations and conservation legislation of variational difficulties. The expository language used this is essentially that of undergraduate calculus instead of differential geometry, making the subject extra obtainable to a scholar viewers. extra refined rules from differential topology and Lie concept are defined from scratch utilizing illustrative examples and workouts. This ebook is perfect for graduate scholars and researchers operating in differential equations, symbolic computation, functions of Lie teams and, to a lesser volume, differential geometry.

Show description

Read or Download A Practical Guide to the Invariant Calculus (Cambridge Monographs on Applied and Computational Mathematics, Vol. 26) PDF

Similar topology books

Category Theory: Proceedings of the International Conference Held in Como, Italy, July 22-28, 1990

With one exception, those papers are unique and entirely refereed learn articles on a variety of functions of type conception to Algebraic Topology, common sense and desktop technology. The exception is a phenomenal and long survey paper via Joyal/Street (80 pp) on a transforming into topic: it supplies an account of classical Tannaka duality in this type of manner as to be available to the overall mathematical reader, and to supply a key for access to extra fresh advancements and quantum teams.

A Mathematical Gift I, II, III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 1-3)

This three-volume set addresses the interaction among topology, features, geometry, and algebra. Bringing the wonder and enjoyable of arithmetic to the school room, the authors provide critical arithmetic in a full of life, reader-friendly type. incorporated are routines and plenty of figures illustrating the most strategies.

Topological Nonlinear Analysis II: Degree, Singularity, and Variations (Progress in Nonlinear Differential Equations and Their Applications Series, Vol. 27)

The most objective of the current quantity is to provide a survey of a few of the main major achievements got through topological equipment in nonlin­ ear research over the last 3 many years. it's meant, no less than partially, as a continuation of Topological Nonlinear research: measure, Singularity and Varia­ tions, released in 1995.

Extra resources for A Practical Guide to the Invariant Calculus (Cambridge Monographs on Applied and Computational Mathematics, Vol. 26)

Sample text

This quantity is in fact the Euclidean curvature of the path x → (x, y(x)) in the plane. 32) is invariant under the prolonged action. This expression is known as the Schwarzian derivative of u with respect to x and is often denoted {u; x}. More generally, we are concerned with q smooth functions uα that depend on p variables xi . g. uα1112222 = ∂7 uα ∂x13 ∂x24 or uβxxyyy = ∂5 uβ . ∂x 2 ∂y 3 We consider these derivative functions as functionally independent coordinates of a so-called jet space, denoted J (X × U ), or J for short, where X is the space whose coordinates are the independent variables, and U the space whose coordinates are the dependent variables.

What if that point is the origin? What is the isotropy group of a circle in the plane? What is the global isotropy group of a circle in the plane? 4 A discrete subgroup of a Lie group G is a subgroup which, as a set, consists of isolated points in G. 5 (Free and effective actions) A group action on M is said to be: free if Gz = {e}, for all z ∈ M locally free if Gz is a discrete subgroup of G, for all z ∈ M effective if G∗M = {e} locally effective if G∗U is a discrete subgroup of G on subsets for every open subset U ⊂ M.

12 Consider the action of the Euclidean group SE(2) on the (u, v) plane given by, u v = cos θ sin θ − sin θ cos θ u v a b + . Assume u = u(x, t) and v = v(x, t). Since x and t are both invariant, we have Dx = Dx , Dt = Dt and hence the prolonged action is g• uK vK cos θ sin θ = − sin θ cos θ uK vK . 37) where (a, b, c) are the coordinates of g ∈ SL(2) near the identity e = (1, 0, 0), we have courtesy of the chain rule that Dx Dt a + bux but = 0 1 Dx Dt and thus Dx = 1 Dx , a + bux D t = Dt − but Dx .

Download PDF sample

Rated 4.94 of 5 – based on 5 votes