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Surfaces, Deformation of. --- Surfaces, Deformation of.

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Surfaces, Deformation of. --- Deformation of surfaces

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The surface characterizations of engineering materials effects their scratch/abrasion/Mar resistance, coating adhesion/strength, and abrasive wear mechanism. Scratching of Materials and Applications has chapters devoted to direct industrial application and contains some of the important works that are being conducted. Scratch testing of materials has grown extensively since the earlier days of the Mohs Scale for ranking minerals according to their relative scratch resistance. This test has been used on metals, ceramics, glasses, polymers and coatings of various types and thickness

Polymers --- Coatings. --- Surfaces, Deformation of. --- Surfaces. --- Deformation of surfaces --- Surface coatings --- Materials --- Surfaces (Technology) --- Coating processes --- Thin films

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In this monograph the authors redevelop the theory systematically using two different approaches. A general mechanism for the deformation of structures on manifolds was developed by Donald Spencer ten years ago. A new version of that theory, based on the differential calculus in the analytic spaces of Grothendieck, was recently given by B. Malgrange. The first approach adopts Malgrange's idea in defining jet sheaves and linear operators, although the brackets and the non-linear theory arc treated in an essentially different manner. The second approach is based on the theory of derivations, and its relationship to the first is clearly explained. The introduction describes examples of Lie equations and known integrability theorems, and gives applications of the theory to be developed in the following chapters and in the subsequent volume.

Differential geometry. Global analysis --- Lie groups --- Lie algebras --- Differential equations --- Groupes de Lie --- Algèbres de Lie --- Equations différentielles --- 514.76 --- Groups, Lie --- Symmetric spaces --- Topological groups --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Equations, Differential --- Bessel functions --- Calculus --- Geometry of differentiable manifolds and of their submanifolds --- Differential equations. --- Lie algebras. --- Lie groups. --- 517.91 Differential equations --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Algèbres de Lie --- Equations différentielles --- 517.91. --- Numerical solutions --- Surfaces, Deformation of --- Surfaces (mathématiques) --- Déformation --- Pseudogroups. --- Pseudogroupes (mathématiques) --- 517.91 --- Adjoint representation. --- Adjoint. --- Affine transformation. --- Alexander Grothendieck. --- Analytic function. --- Associative algebra. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Bundle map. --- Category of topological spaces. --- Cauchy–Riemann equations. --- Coefficient. --- Commutative diagram. --- Commutator. --- Complex conjugate. --- Complex group. --- Complex manifold. --- Computation. --- Conformal map. --- Continuous function. --- Coordinate system. --- Corollary. --- Cotangent bundle. --- Curvature tensor. --- Deformation theory. --- Derivative. --- Diagonal. --- Diffeomorphism. --- Differentiable function. --- Differential form. --- Differential operator. --- Differential structure. --- Direct proof. --- Direct sum. --- Ellipse. --- Endomorphism. --- Equation. --- Exact sequence. --- Exactness. --- Existential quantification. --- Exponential function. --- Exponential map (Riemannian geometry). --- Exterior derivative. --- Fiber bundle. --- Fibration. --- Frame bundle. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- Group isomorphism. --- Groupoid. --- Holomorphic function. --- Homeomorphism. --- Integer. --- J-invariant. --- Jacobian matrix and determinant. --- Jet bundle. --- Linear combination. --- Linear map. --- Manifold. --- Maximal ideal. --- Model category. --- Morphism. --- Nonlinear system. --- Open set. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Pointwise. --- Presheaf (category theory). --- Pseudo-differential operator. --- Pseudogroup. --- Quantity. --- Regular map (graph theory). --- Requirement. --- Riemann surface. --- Right inverse. --- Scalar multiplication. --- Sheaf (mathematics). --- Special case. --- Structure tensor. --- Subalgebra. --- Subcategory. --- Subgroup. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Tensor field. --- Tensor product. --- Theorem. --- Torsion tensor. --- Transpose. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume element. --- Surfaces (mathématiques) --- Déformation --- Analyse sur une variété