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The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.
The genus (sometimes called the demigenus or Euler genus) of a connected non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ , via the relationship χ = 2 − g , where g is the non-orientable ...
File:Lagrangian vs Eulerian [further explanation needed] Eulerian perspective of fluid velocity versus Lagrangian depiction of strain. In classical field theories , the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time.
For example, the teardrop orbifold has Euler characteristic 1 + 1 / p , where p is a prime number corresponding to the cone angle 2 π / p . The concept of Euler characteristic of the reduced homology of a bounded finite poset is another generalization, important in combinatorics. A poset is "bounded" if it has smallest and ...
For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves.
An example of a set of equations written in conservation form are the Euler equations of fluid flow: + = + (+) = + ((+)) = Each of these represents the conservation of mass , momentum and energy , respectively.
A tabulation of the numbers in a triangular array is called the Euler triangle or Euler's triangle. It shares some common characteristics with Pascal's triangle . Values of A ( n , k ) {\textstyle A(n,k)} (sequence A008292 in the OEIS ) for 0 ≤ n ≤ 9 {\textstyle 0\leq n\leq 9} are:
Euler invented the calculus of variations including its most well-known result, the Euler–Lagrange equation. Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory.