Category: Applied mathematics

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1221 days ago

The function f is a rule that assigns an incoming number x, a uniquely defined outgoing number y. y = f(x)

1221 days ago

Real scalar functions of matrix argument, when the matrices are real, will be dealt with. It is diﬃcult to develop a theory of functions of matrix argument for general matrices.

1221 days ago

Start with sinx. It has period 2 π since sin(x +2π) = sinx. It is an odd function since sin(−x)=−sinx, and it vanishes at x = 0 andx = π. Every function sinnx has those three properties, and Fourier looked at inﬁnite combinations of the sines

1221 days ago

As a set, C = R2 = {(x,y)| x,y ∈ R}. In other words, elements of C are pairs of real numbers.

1221 days ago

When reading or writing mathematics you should always remember that the mathematical symbols which are used are simply abbreviations for words. Mechanically replacing the symbols by the words they represent should result in grammatically correct and complete sentences.

1221 days ago

Given a vector space, V , an inner product on V is a rule for multiplying elements of V together so that the result is a scalar. If u and v are vectors in V , then their inner product is written hu,vi

1221 days ago

Vector spaces owe their importance to the fact that so many models arising in the solutions of speciﬁc problems turn out to be vector spaces. For this reason the basic concepts introduced in them have a certain universality and are ones we encounter, and keep encountering, in so many diverse contexts.

1221 days ago

Linear Algebra, which is one of the most widely used mathematical theories around. Linear Algebra ﬁnds applications in virtually every area of mathematics, including Multivariate Calculus, Diﬀerential Equations, and Probability Theory.

1221 days ago

Body segment rotations combine to produce linear motion of the whole body or of a specific point on a body segment or implement.

1221 days ago

Flows in conduits or channels are of interest in science, engineering, and everyday life. Flows in closed conduits or channels, like pipes or air ducts, are entirely in contact with rigid boundaries. Most closed conduits in engineering applications are either circular or rectangular in cross section.

1221 days ago

If there is no force on a mass, acceleration is zero m˙ v = 0 =⇒mv is constant

1221 days ago

Forces acting of rigid bodies can be also separated in two groups: (a) The external forces, represent the action of other bodies on the rigid body under consideration; (b) The internal forcesare the forces which hold together the particles forming the rigid body. Only external forces can impart to the rigid body a motion of translation or rotation or both

1221 days ago

Riemann had revolutionized the fields of analysis, geometry and mathematical physics. His ideas concerning geometry of space had a profound effect on the development of modern theoretical physics.

1221 days ago

We deﬁnedRb a f(t)dt under the conditions that f is deﬁned and bounded on the bounded interval [a,b].

1221 days ago

A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X.

1221 days ago

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. →A function is continuous at a point if the limit is the same as the value of the function.

1221 days ago

Open formulas - use interior points only. Extended formulas – piecewise sum of integration formula.

1221 days ago

•Calculus is the mathematics of change. •Engineers must continuously deal with systems and processes that change, making calculus an essential tool of our profession. •At the heart of calculus are the related mathematical concepts of differentiation and integration.

1221 days ago

The ﬁnite diﬀerence approximations for derivatives are one of the simplest and of the oldest methods to solve diﬀerential equations.

1221 days ago

An equation relating a function to its derivatives of a single variable(in such a way that the function itself can be determined)

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A ring is a triple (R, +, ·) where R is a set, and + and · are binary operations on R (called addition and multiplication respectively) so that: (1) (R,+) is an abelian group (with identity denoted by 0 and the inverse of x é R denoted by -x, as usual.) (2) Multiplication is associative. (3) The following distributive laws hold "x, y, z é R: x(y+z) = xy + xz (left distributive law) (x+y)z = xz + yz (right distributive law)

1221 days ago

Graph homomorphism maps adjacent vertices to adjacent vertices between two graphs.

1221 days ago

An integral domain R is said to be Euclidean if there exists a map φ: R\{0} → N such that given any a,b ∈ R, there exist q and r such that a = bq+r with either r = 0 or φ(r) < φ(b). Any such ring is a principal ideal domain (PID).

1221 days ago

A group is a set G which is equipped with an operation ∗ and a special element e ∈ G, called the identity.

1221 days ago