By Daniel Martin
Read Online or Download Manifold Theory: An Introduction for Mathematical Physicists PDF
Similar mathematical physics books
Gauge Symmetries and Fibre Bundles
A thought outlined by means of an motion that is invariant below a time based workforce of alterations should be referred to as a gauge conception. popular examples of such theories are these outlined by way of the Maxwell and Yang-Mills Lagrangians. it's broadly believed these days that the basic legislation of physics need to be formulated when it comes to gauge theories.
Mathematical Methods Of Classical Mechanics
During this textual content, the writer constructs the mathematical gear of classical mechanics from the start, studying all of the easy difficulties in dynamics, together with the idea of oscillations, the idea of inflexible physique movement, and the Hamiltonian formalism. this contemporary approch, in response to the speculation of the geometry of manifolds, distinguishes iteself from the conventional method of normal textbooks.
- Mathematische Methoden der Physik I: Analysis
- Rigid Body Mechanics: Mathematics, Physics and Applications (Physics Textbook)
- Partial Differential Equations of Mathematical Physics
- Idempotent Mathematics And Mathematical Physics: International Workshop, February 3-10, 2003, Erwin Schrodinger International Institute For ... Vienna, Austria
- Statistical physics and field theory
Extra info for Manifold Theory: An Introduction for Mathematical Physicists
Sample text
E*,®.. ® e ® e ' ' ® . l(ii). Hence Γ/· · · / ' c, ® . ®e, ®e'<®.. ®e>' = Γ/' /- e, ® . ®e, ®e>>®.. ®e'' and so a unique tensor is defined by the given set of numbers and their law of transformation. Π Sec. 1] Tensors 33 Some special cases of the last theorem are these: (i) (ii) (iii) (iv) (ν) For a contravariant vector u, u'" = A ·" u'. For a covariant vector ω, ω,. = A •· ω,. For a contravariant tensor Τ of order 2, Τ = A /' Λ/' Γ'Λ For a covariant tensor Τ of order 2, Γ,γ = A j.
10 31 For a change of basis from (e,) to (ey), where ey = A [• e , we have that x' = A' x'' and hence that A$. =A\. =Al = l and that all remaining coefficients are zero. Then ( To. = r i4. A{. 0 Tj = A S ' A?. TQO + AJ}. A\. T 0i 2 = cosh α sinh α T w r +A& A° T y 2 + cosh aT + sinh ar, + cosha sinha 01 0 + A^. A\. l0 T n and T. l0 2 = cosh α sinh α Τ ω 2 + sinh aT , + cosh aT + cosh a sinh a T „ 0 10 T u Sec. 1] Tensors 39 Hence TOT — ΊΊ·ο· = ΤΌι - T* . 10 (ii) is proved similarly. Symmetric and skew-symmetric tensors A contravariant tensor Τ of order 2 is said to be symmetric if Τ(ω, σ) = Τ(σ, ω) for all 1-forms (covectors) ω and σ.
Q We are now ready to define the Hodge star operator. Let ω be a fixed p-form and σ any (n-p)-form. Then the mapping from A" '(i/,) A"(V„), defined by σι-»ωΛσ, is linear and we may write - / t o ωΛσ=/(ω,σ) e'A.. Ae" , where (e') is the basis dual to the basis (e ) of V„, and/(ω, o)e R is linear in σ. 1 applied to the vector space A" '(V„) there exists a unique element *a>of A"-''(V„) such that/ϊω,σ) = (*ω, σ). Thus t _/ ω Λ σ = (*ω,σ) e'A.. Ae (2) The mapping: A''(V„)-> A"-''(K„) taking ω to *ω as defined by (2) is called the Hodge star operator and *ω is called the dual of ω.