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  2. Closed and exact differential forms - Wikipedia

    en.wikipedia.org/wiki/Closed_and_exact...

    In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.

  3. Vector notation - Wikipedia

    en.wikipedia.org/wiki/Vector_notation

    A scalar beside a vector (either or both of which may be in parentheses) implies scalar multiplication. The two common operators, a dot and a rotated cross, are also acceptable (although the rotated cross is almost never used), but they risk confusion with dot products and cross products, which operate on two vectors.

  4. First fundamental form - Wikipedia

    en.wikipedia.org/wiki/First_fundamental_form

    The first fundamental form completely describes the metric properties of a surface. Thus, it enables one to calculate the lengths of curves on the surface and the areas of regions on the surface. The line element ds may be expressed in terms of the coefficients of the first fundamental form as d s 2 = E d u 2 + 2 F d u d v + G d v 2 ...

  5. Differential geometry of surfaces - Wikipedia

    en.wikipedia.org/wiki/Differential_geometry_of...

    Other weakened forms of regular surfaces occur in computer-aided design, where a surface is broken apart into disjoint pieces, with the derivatives of local parametrizations failing to even be continuous along the boundaries. [citation needed] A hyperboloid of two sheets A torus A helicoid. Simple examples.

  6. Scalar multiplication - Wikipedia

    en.wikipedia.org/wiki/Scalar_multiplication

    Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications −a and 2a of a vector a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra [1] [2] [3] (or more generally, a module in abstract algebra [4] [5]).

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