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This is because the surface N has a unique class of one-sided curves such that, when N is cut open along such a curve C, the resulting surface is a torus with a disk removed. As an unoriented surface, its mapping class group is GL ( 2 , Z ) {\displaystyle \operatorname {GL} (2,\mathbb {Z} )} .
The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x. In other words, the area of this "strip" would be A(x + h) − A(x). There is another way to estimate the area of this same strip. As shown in the accompanying figure, h is multiplied by f(x) to find ...
In general a curve will not have 4th-order contact with any circle. However, 4th-order contact can occur generically in a 1-parameter family of curves, at a curve in the family where (as the parameter varies) two vertices (one maximum and one minimum) come together and annihilate. At such points the second derivative of curvature will be zero.
If f(x) represents speed as it varies over time, the distance traveled between the times represented by a and b is the area of the region between f(x) and the x-axis, between x = a and x = b. To approximate that area, an intuitive method would be to divide up the distance between a and b into several equal segments, the length of each segment ...
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. This curve has total curvature 6π, and turning number 3.. The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if ...
Surfaces of revolution give another important class of surfaces that can be easily parametrized. If the graph z = f ( x ) , a ≤ x ≤ b is rotated about the z -axis then the resulting surface has a parametrization r ( u , ϕ ) = ( u cos ϕ , u sin ϕ , f ( u ) ) , a ≤ u ≤ b , 0 ≤ ϕ < 2 π . {\displaystyle \mathbf {r} (u,\phi ...
Total variation distance is half the absolute area between the two curves: Half the shaded area above. In probability theory, the total variation distance is a statistical distance between probability distributions, and is sometimes called the statistical distance, statistical difference or variational distance.
The word "class" in the term "equivalence class" may generally be considered as a synonym of "set", although some equivalence classes are not sets but proper classes. For example, "being isomorphic " is an equivalence relation on groups , and the equivalence classes, called isomorphism classes , are not sets.