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  2. Hypersurface - Wikipedia

    en.wikipedia.org/wiki/Hypersurface

    In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.A hypersurface is a manifold or an algebraic variety of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. [1]

  3. Level set - Wikipedia

    en.wikipedia.org/wiki/Level_set

    When n = 3, a level set is called a level surface (or isosurface); so a level surface is the set of all real-valued roots of an equation in three variables x 1, x 2 and x 3. For higher values of n, the level set is a level hypersurface, the set of all real-valued roots of an equation in n > 3 variables. A level set is a special case of a fiber.

  4. Hyperplane - Wikipedia

    en.wikipedia.org/wiki/Hyperplane

    In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V.The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can ...

  5. Laplace–Beltrami operator - Wikipedia

    en.wikipedia.org/wiki/Laplace–Beltrami_operator

    More generally, one can formulate a similar trick using the normal bundle to define the Laplace–Beltrami operator of any Riemannian manifold isometrically embedded as a hypersurface of Euclidean space. One can also give an intrinsic description of the Laplace–Beltrami operator on the sphere in a normal coordinate system.

  6. Gauss–Codazzi equations - Wikipedia

    en.wikipedia.org/wiki/Gauss–Codazzi_equations

    In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.

  7. Second fundamental form - Wikipedia

    en.wikipedia.org/wiki/Second_fundamental_form

    The second fundamental form of a parametric surface S in R 3 was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin. Then f and its partial derivatives with respect to x and y vanish at (0,0).

  8. Tarski's axioms - Wikipedia

    en.wikipedia.org/wiki/Tarski's_axioms

    Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity (i.e. is formulable as an elementary theory). As such, it does not require an underlying set theory. The only primitive objects of the system are "points" and the only primitive ...

  9. Quadric (algebraic geometry) - Wikipedia

    en.wikipedia.org/wiki/Quadric_(algebraic_geometry)

    In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. An example is the quadric surface =