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A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K.The Clifford algebra Cl(V, Q) is the "freest" unital associative algebra generated by V subject to the condition [c] = , where the product on the left is that of the algebra, and the 1 on the right is the algebra's ...
Difficult integrals may also be solved by simplifying the integral using a change of variables given by the corresponding Jacobian matrix and determinant. [1] Using the Jacobian determinant and the corresponding change of variable that it gives is the basis of coordinate systems such as polar, cylindrical, and spherical coordinate systems.
The ESV Study Bible was first published in October 2008, having been supported by a $1 million campaign. [6] The first printing of the ESVSB, consisting of 100,000 copies, sold out prior to completion. Within the first six months of availability, 300,000 copies had been printed in total. [1] The ESVSB eventually went on to sell over 1 million ...
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
By September 2024, the ESV Study Bible had sold more than 2.5 million copies. [35] ESV New Classic Reference Bible (Commemorative Edition; top grain leather) In 2011, Crossway published a special limited edition ESV New Classic Reference Bible to commemorate the 400th anniversary of the King James Version (KJV) first being published. [36]
[4] [5] The problem was discussed in a number of publications. [2] [3] [4] In 1960, the equation was among the questions on the William Lowell Putnam Competition, [6] [7] which prompted Alvin Hausner to extend results to algebraic number fields. [3] [8]
End-systolic volume (ESV) is the volume of blood in a ventricle at the end of contraction, or systole, and the beginning of filling, or diastole. ESV is the lowest volume of blood in the ventricle at any point in the cardiac cycle. The main factors that affect the end-systolic volume are afterload and the contractility of the heart.
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials.It is commonly used to solve non-exact ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be ...