Definite quadratic form - Wikipedia

Definite quadratic form - Wikipedia: "the quadratic form is called positive definite or negative definite."



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Riemannian manifold - Wikipedia

Riemannian manifold - Wikipedia: " real, smooth manifold M equipped with an inner product {\displaystyle g_{p}} on the tangent space {\displaystyle T_{p}M} at each point {\displaystyle p} that varies smoothly from point to point "



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Lie Groups (why do mirrors reverse left and right, but not up and down?)

Lie Groups: "Digression: the well-known puzzle, ``why do mirrors reverse left and right, but not up and down?'' is resolved mathematically by pointing out that a mirror perpendicular to the y-axis performs the reflection:

"



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Group action - Wikipedia

Group action - Wikipedia: "stabilizer subgroup of G with respect to x (also called the isotropy group "



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Differential of a function - Wikipedia

Differential of a function - Wikipedia: "The differential dy is defined by

{\displaystyle dy=f'(x)\,dx,}
where {\displaystyle f'(x)} is the derivative of f with respect to x, and dx is an additional real variable (so that dy is a function of x and dx). "



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What first-order linearity really is

‘to first order everything is linear’. This is tautological, of course, but it really says
that most functions are differentiable and we can approximate by the first derivative.



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Differential (infinitesimal) - Wikipedia

Differential (infinitesimal) - Wikipedia: "the differential dy of y is related to dx by the formula

{\displaystyle \mathrm {d} y={\frac {\mathrm {d} y}{\mathrm {d} x}}\,\mathrm {d} x,}
where dy/dx denotes the derivative of y with respect to x."



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