Search results
Results from the WOW.Com Content Network
The convolution of f and g exists if f and g are both Lebesgue integrable functions in L 1 (R d), and in this case f∗g is also integrable (Stein & Weiss 1971, Theorem 1.3). This is a consequence of Tonelli's theorem. This is also true for functions in L 1, under the discrete convolution, or more generally for the convolution on any group.
Thus, it enables one to calculate the lengths of curves on the surface and the areas of regions on the surface. The line element ds may be expressed in terms of the coefficients of the first fundamental form as d s 2 = E d u 2 + 2 F d u d v + G d v 2 . {\displaystyle ds^{2}=E\,du^{2}+2F\,du\,dv+G\,dv^{2}\,.}
The area between the graphs of two functions is equal to the integral of one function, f(x), minus the integral of the other function, g(x): A = ∫ a b ( f ( x ) − g ( x ) ) d x , {\displaystyle A=\int _{a}^{b}(f(x)-g(x))\,dx,} where f ( x ) {\displaystyle f(x)} is the curve with the greater y-value.
A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with ...
The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics—sometimes infinitesimally small) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together.
It may be used to prove Nicomachus's theorem that the sum of the first cubes equals the square of the sum of the first positive integers. [2] Summation by parts is frequently used to prove Abel's theorem and Dirichlet's test.
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is T = b h / 2 , {\displaystyle T=bh/2,} where b is the length of the base of the triangle, and h is the height or altitude of the triangle.