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The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with the work of Riemann , the intrinsic point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way.
The Jacobian matrix represents the differential of f at every point where f is differentiable. In detail, if h is a displacement vector represented by a column matrix , the matrix product J ( x ) ⋅ h is another displacement vector, that is the best linear approximation of the change of f in a neighborhood of x , if f ( x ) is differentiable ...
See also multivariable calculus, list of multivariable calculus topics. Manifold. Differentiable manifold; Smooth manifold; Banach manifold; Fréchet manifold; Tensor analysis. Tangent vector
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The differential-geometric properties of a parametric curve (such as its length, its Frenet frame, and its generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class itself. The equivalence classes are called C r-curves and are central objects studied in the differential geometry of curves.
Calibrated geometry; Cartan connection; Cartan's equivalence method; Catalan's minimal surface; Caustic (mathematics) Cayley's ruled cubic surface; Center of curvature; Chentsov's theorem; Chern–Simons form; Chern–Weil homomorphism; Chern's conjecture (affine geometry) Chern's conjecture for hypersurfaces in spheres; Clairaut's relation ...
In the language of differential geometry, this derivative is a one-form on the punctured plane. It is closed (its exterior derivative is zero) but not exact , meaning that it is not the derivative of a 0-form (that is, a function): the angle θ {\\displaystyle \\theta } is not a globally defined smooth function on the entire punctured plane.
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, [1] is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
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