By P. Grisvard
This ebook experiences the suggestions of a boundary challenge close to nook edges and vertices. The exposition is introductory and self-contained. It makes a speciality of real-life difficulties thought of within the real geometry met within the functions. The e-book highlights the singular recommendations which hold the most actual details and that are given of their so much explicitform to assist capability clients.
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Additional resources for Singularities in boundary value problems (Recherches en mathématiques appliquées)
Example 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.