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Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities; List of volume formulas – Quantity of three-dimensional space
This is because the n-dimensional dV element is in general a parallelepiped in the new coordinate system, and the n-volume of a parallelepiped is the determinant of its edge vectors. The Jacobian can also be used to determine the stability of equilibria for systems of differential equations by approximating behavior near an equilibrium point.
The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation:
Some SI units of volume to scale and approximate corresponding mass of water. To ease calculations, a unit of volume is equal to the volume occupied by a unit cube (with a side length of one). Because the volume occupies three dimensions, if the metre (m) is chosen as a unit of length, the corresponding unit of volume is the cubic metre (m 3).
Taking an example, the area under the curve y = x 2 over [0, 2] can be procedurally computed using Riemann's method. The interval [0, 2] is firstly divided into n subintervals, each of which is given a width of 2 n {\displaystyle {\tfrac {2}{n}}} ; these are the widths of the Riemann rectangles (hereafter "boxes").
Consider the linear subspace of the n-dimensional Euclidean space R n that is spanned by a collection of linearly independent vectors , …,. To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the : (), = ….
A three-dimensional rectangular wire frame that is twisted can take the shape of a bow tie. The interior of a crossed rectangle can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise. A crossed rectangle may be considered equiangular if right and left turns are allowed.
The perimeter of a stadium is calculated by the formula = (+) where a is the length of the straight sides and r is the radius of the semicircles. With the same parameters, the area of the stadium is A = π r 2 + 2 r a = r ( π r + 2 a ) {\displaystyle A=\pi r^{2}+2ra=r(\pi r+2a)} .