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The volume of a tetrahedron can be obtained in many ways. It can be given by using the formula of the pyramid's volume: =. where is the base' area and is the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apices to the opposite faces are inversely proportional to the areas of ...
This angle may be calculated from the dot product of the two vectors, defined as a ⋅ b = ‖ a ‖ ‖ b ‖ cos θ where ‖ a ‖ denotes the length of vector a. As shown in the diagram, the dot product here is –1 and the length of each vector is √ 3, so that cos θ = – 1 / 3 and the tetrahedral bond angle θ = arccos ...
In electromagnetism, Maxwell's equations can be written using multiple integrals to calculate the total magnetic and electric fields. [12] In the following example, the electric field produced by a distribution of charges given by the volume charge density ρ( r →) is obtained by a triple integral of a vector function:
as can be seen by multiplying the previous formula by x n+1, to get the volume under the n-simplex as a function of its vertex distance x from the origin, differentiating with respect to x, at = / (where the n-simplex side length is 1), and normalizing by the length / + of the increment, (/ (+), …, / (+)), along the normal vector.
Area#Area formulas – Size of a two-dimensional surface; 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
Expressing the Navier–Stokes vector equation in Cartesian coordinates is quite straightforward and not much influenced by the number of dimensions of the euclidean space employed, and this is the case also for the first-order terms (like the variation and convection ones) also in non-cartesian orthogonal coordinate systems.
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:
In mathematics (particularly multivariable calculus), a volume integral (∭) is an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities, or to calculate mass from a corresponding density ...