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The dot planimeter is physical device for estimating the area of shapes based on the same principle. It consists of a square grid of dots, printed on a transparent sheet; the area of a shape can be estimated as the product of the number of dots in the shape with the area of a grid square. [8]
If one wants to extend the natural functional calculus for polynomials on the spectrum of an element of a Banach algebra to a functional calculus for continuous functions (()) on the spectrum, it seems obvious to approximate a continuous function by polynomials according to the Stone-Weierstrass theorem, to insert the element into these polynomials and to show that this sequence of elements ...
the sinc-function becomes a continuous function on all real numbers. The term removable singularity is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points. A more involved construction of continuous functions is the function composition.
Let X denote the real numbers ℝ with the usual Euclidean topology and let Y denote ℝ with the indiscrete topology (where note that Y is not Hausdorff and that every function valued in Y is continuous). Let f : X → Y be defined by f(0) = 1 and f(x) = 0 for all x ≠ 0. Then f : X → Y is continuous but its graph is not closed in X × Y. [4]
Clearly the radius will not depend on the position γ(t), only on the velocity γ′(t) and acceleration γ″(t). There are only three independent scalars that can be obtained from two vectors v and w, namely v · v, v · w, and w · w. Thus the radius of curvature must be a function of the three scalars | γ′(t) | 2, | γ″(t) | 2 and γ ...
Consider for example the function (,) = (+,) which maps every point of the open unit disk to another point on the open unit disk to the right of the given one. But for the closed unit disk it fixes every point on the half circle x 2 + y 2 = 1 , x > 0. {\displaystyle x^{2}+y^{2}=1,x>0.}
Then, we use the so-called keyhole contour, which consists of a small circle about the origin of radius ε say, extending to a line segment parallel and close to the positive real axis but not touching it, to an almost full circle, returning to a line segment parallel, close, and below the positive real axis in the negative sense, returning to ...
The cross-section's area is therefore the area of the larger circle minus the area of the smaller circle: () = (()) = (()). The radius R does not appear in the last quantity. Therefore, the area of the horizontal cross-section at height y {\displaystyle y} does not depend on R {\displaystyle R} , as long as y ≤ h 2 ≤ R {\displaystyle y\leq ...