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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. Quadratic form (statistics) - Wikipedia

   en.wikipedia.org/wiki/Quadratic_form_(statistics)

   Since the quadratic form is a scalar quantity, = ⁡ (). Next, by the cyclic property of the trace operator, ⁡ [⁡ ()] = ⁡ [⁡ ()]. Since the trace operator is a linear combination of the components of the matrix, it therefore follows from the linearity of the expectation operator that

4. Gradient theorem - Wikipedia

   en.wikipedia.org/wiki/Gradient_theorem

   The converse statement of the gradient theorem also has a powerful generalization in terms of differential forms on manifolds. In particular, suppose ω is a form defined on a contractible domain, and the integral of ω over any closed manifold is zero. Then there exists a form ψ such that ω = dψ.

5. Second fundamental form - Wikipedia

   en.wikipedia.org/wiki/Second_fundamental_form

   and the second fundamental form at the origin in the coordinates (x,y) is the quadratic form L d x 2 + 2 M d x d y + N d y 2 . {\displaystyle L\,dx^{2}+2M\,dx\,dy+N\,dy^{2}\,.} For a smooth point P on S , one can choose the coordinate system so that the plane z = 0 is tangent to S at P , and define the second fundamental form in the same way.

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   You've Got Mail!® Millions of people around the world use AOL Mail, and there are times you'll have questions about using it or want to learn more about its features. That's why AOL Mail Help is here with articles, FAQs, tutorials, our AOL virtual chat assistant and live agent support options to get your questions answered.