terminology - What are the differences between rings, groups, and fields? - Mathematics Stack Exchange

terminology - What are the differences between rings, groups, and fields? - Mathematics Stack Exchange: "They should feel similar! In fact, every ring is a group, and every field is a ring. A ring is a group with an additional operation, where the second operation is associative and the distributive properties make the two operations "compatible".

A field is a ring such that the second operation also satisfies all the group properties (after throwing out the additive identity); i.e. it has multiplicative inverses, multiplicative identity, and is commutative."



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Field extension - Wikipedia

Field extension - Wikipedia: "Q(√2) = {a + b√2 | a, b ∈ Q} is the smallest extension of Q that includes every real solution to the equation x2 = 2."



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Steel square - Wikipedia

Steel square - Wikipedia: "framing square. "



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Homogeneous polynomial - Wikipedia

Homogeneous polynomial - Wikipedia: " is not homogeneous, because the sum of exponents does not match from term to term. A polynomial is homogeneous if and only if it defines a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial.[2] A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis.
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Differentiable manifold - Wikipedia

Differentiable manifold - Wikipedia: "that is locally homeomorphic to a linear space, by a collection (called an atlas) of homeomorphisms called charts. "



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Derivative - Wikipedia

Derivative - Wikipedia: "Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold M is a space that can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R3. The derivative (or differential) of a (differentiable) map f: M → N between manifolds, at a point x in M, is then a linear map from the tangent space of M at x to the tangent space of N at f(x). The derivative function becomes a map between the tangent bundles of M and N. This definition is fundamental in differential geometry and has many uses – see pushforward (differential) and pullback (differential geometry).
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Exterior derivative - Wikipedia

Exterior derivative - Wikipedia: "so d( f ∗ω) =  f ∗dω, where  f ∗ denotes the pullback of  f . This follows from that  f ∗ω(·), by definition, is ω( f∗(·)),  f∗ being the pushforward of  f . Thus d is a natural transformation from Ωk to Ωk+1.
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