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In algebraic number theory, a quadratic field is an algebraic number field of degree two over , the rational numbers. Every such quadratic field is some Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}})} where d {\displaystyle d} is a (uniquely defined) square-free integer different from 0 {\displaystyle 0} and 1 {\displaystyle 1} .
A quadric is said to be non-degenerate if the matrix is invertible. A non-degenerate quadric is non-singular in the sense that its projective completion has no singular point (a cylinder is non-singular in the affine space, but it is a degenerate quadric that has a singular point at infinity).
The two families of lines on a smooth (split) quadric surface. In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine ...
The strain rate tensor typically varies with position and time within the material, and is therefore a (time-varying) tensor field. It only describes the local rate of deformation to first order ; but that is generally sufficient for most purposes, even when the viscosity of the material is highly non-linear.
Stress analysis is specifically concerned with solid objects. The study of stresses in liquids and gases is the subject of fluid mechanics.. Stress analysis adopts the macroscopic view of materials characteristic of continuum mechanics, namely that all properties of materials are homogeneous at small enough scales.
One of the basic examples of norms comes from quadratic field extensions () / where is a square-free integer.. Then, the multiplication map by on an element + is (+) = +.The element + can be represented by the vector
Americans don’t know the full extent of what Elon Musk is doing as he embeds alongside President Donald Trump at the top of the federal government.
A tangential vector field X on S assigns, to each p in S, a tangent vector X p to S at p. According to the "intrinsic" definition of tangent vectors given above, a tangential vector field X then assigns, to each local parametrization f : V → S , two real-valued functions X 1 and X 2 on V , so that