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In vector calculus, the Jacobian matrix (/ dʒəˈkoʊbiən /, [1][2][3] / dʒɪ -, jɪ -/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output ...
A vector v (red) represented by • a vector basis (yellow, left: e 1, e 2, e 3), tangent vectors to coordinate curves (black) and • a covector basis or cobasis (blue, right: e 1, e 2, e 3), normal vectors to coordinate surfaces (grey) in general (not necessarily orthogonal) curvilinear coordinates (q 1, q 2, q 3). The basis and cobasis do ...
This is a method for removing surds from expressions (or at least moving them), applying to division by some combinations involving square roots. For example: The denominator of 5 3 + 4 {\displaystyle {\dfrac {5}{{\sqrt {3}}+4}}} can be rationalised as follows:
A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as a / b or , where a and b are both integers. [9] As with other fractions, the denominator (b) cannot be zero. Examples include 1 2 , − 8 5 , −8 5 , and 8 −5 .
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2] Since the problem had withstood the attacks of ...
In mathematics, the definite integral. is the area of the region in the xy -plane bounded by the graph of f, the x -axis, and the lines x = a and x = b, such that area above the x -axis adds to the total, and that below the x -axis subtracts from the total. The fundamental theorem of calculus establishes the relationship between indefinite and ...
Bernoulli's inequality. An illustration of Bernoulli's inequality, with the graphs of and shown in red and blue respectively. Here, In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of . It is often employed in real analysis. It has several useful variants: [1]
The integral of the secant function was one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory. [3] He applied his result to a problem concerning nautical tables. [1] In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums. [4]