first fundamental formの例文

• The first fundamental form of the surface in these coordinates has a special form
• The principal curvatures are the invariants of the pair consisting of the second and first fundamental forms.
• Is called the first fundamental form associated to the metric, while " ds " is the line element.
• The first fundamental form evaluated on this pair of vectors is their dot product, and the angle can be found from the standard formula
• Is a quadratic form on the tangent plane to the surface that, together with the first fundamental form, determines the curvatures of curves on the surface.
• To connect this point of view with the embedded into "'R "'3 and endowed with the Riemannian metric given by the first fundamental form.
• The first fundamental form may be viewed as a family of positive definite symmetric bilinear forms on the tangent plane at each point of the surface depending smoothly on the point.
• Like the first fundamental form, the second fundamental form may be viewed as a family of symmetric bilinear forms on the tangent plane at each point of the surface depending smoothly on the point.
• The classical area element given by dA = | X _ u \ times X _ v | \ du \, dv can be expressed in terms of the first fundamental form with the assistance of Lagrange's identity,
• At any point where f _ x = f _ y = 0, the first fundamental form g _ { ij } of the surface mapping f is the identity matrix and the second fundamental form b _ { ij } is
• A point " p " in a Riemannian submanifold is umbilical if, at " p ", the ( vector-valued ) Second fundamental form is some normal vector tensor the induced metric ( First fundamental form ).
• Although the primary invariant in the study of the intrinsic geometry of surfaces is the metric ( the first fundamental form ) and the Gaussian curvature, certain properties of surfaces also depend on an embedding into ( or a higher dimensional Euclidean space ).
• :: : : : : : . . . and two surfaces are said to be isometric if there is an isometry between them, which is the case iff they have the same first fundamental form . . .-- talk ) 15 : 58, 8 November 2009 ( UTC)
• Quantum gravoelectrodynamics is the field of scientific study concerning the properties of electrodynamics and general relativity viewed through the tools of differential geometry . ?The origins of quantum gravoelectrodynamics can be found in the research of Friedrich Gauss [ 1 ] [ 2 ] and Bernhard Riemann [ 2 ] . ?Their vision was a search to find a theory based on analysis and calculations that explained nature . ?In their work they found many mathematical tools which can be used to understand electrostatics, geometry and spacetime . ?A more modern conceptualization can be found in John Wheeler s " Geometrodynamics . " ?The central idea here is that gravitation and electromagnetism can be described in terms of geometry itself [ 3 ] . ?Quantum gravoelectrodynamics starting point is the Minkowskian idea of spacetime [ 4 ] . ?For general relativity analysis occurs in spacetime and one takes derivatives to derive the matelectric field, the matmagnetic field, mass, velocity, matter waves, matter radiation and the other element of classical analysis . ?The manifold here is the traditional spacetime with units of distance . ?For electrodynamics analysis occurs in pospotential spacetime in which one takes derivatives to derive traditional concepts like the electric field, the magnetic field, charge waves ( fka electromagnetic waves ), charge radiation ( fka electromagnetic radiation ), current and charge . ?The manifold here has units of distance times the square root of Newtons divided by Amperes . ?Tools from differential geometry like the shape operator, the first fundamental form, the second fundamental form and the third fundamental form are used to find normal curvature, principal curvature, Gaussian curvature and mean curvatures.