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Pythagoras could be called the first known string theorist. Pythagoras, an excellent lyre player, figured out the first known string physics -- the harmonic relationship. Pythagoras realized that vibrating Lyre strings of equal tensions but different lengths would produce harmonious notes (i.e. middle C and high C) if the ratio of the lengths of the two strings were a whole number.
Pythagoras discovered this by looking and listening. Today that information is more precisely encoded into mathematics, namely the wave equation for a string with a tension T and a mass per unit length m. If the string is described in coordinates as in the drawing below, where x is the distance along the string and y is the height of the string, as the string oscillates in time t,

then the equation of motion is the one-dimensional wave equation

Nonrelativistic string equation

where vw is the wave velocity along the string.
When solving the equations of motion, we need to know the "boundary conditions" of the string. Let's suppose that the string is fixed at each end and has an unstretched length L. The general solution to this equation can be written as a sum of "normal modes", here labeled by the integer n, such that

sum of string normal modes

The condition for a normal mode is that the wavelength be some integral fraction of twice the string length, or

The frequency of the normal mode is then

The normal modes are what we hear as notes. Notice that the string wave velocity vw increases as the tension of the string is increased, and so the normal frequency of the string increases as well. This is why a guitar string makes a higher note when it is tightened.
But that's for a nonrelativistic string, one with a wave velocity much smaller than the speed of light. How do we write the equation for a relativistic string?
According to Einstein's theory, a relativistic equation has to use coordinates that have the proper Lorentz transformation properties. But then we have a problem, because a string oscillates in space and time, and as it oscillates, it sweeps out a two-dimensional surface in spacetime that we call a world sheet (compared with the world line of a particle).
In the nonrelativistic string, there was a clear difference between the space coordinate along the string, and the time coordinate. But in a relativistic string theory, we wind up having to consider the world sheet of the string as a two-dimensional spacetime of its own, where the division between space and time depends upon the observer.
The classical equation can be written as

Relativistic string eqaution

where s and t are coordinates on the string world sheet representing space and time along the string, and the parameter c2 is the ratio of the string tension to the string mass per unit length.
These equations of motion can be derived from Euler-Lagrange equations from an action based on the string world sheet

Bosonic string action

The spacetime coordinates Xm of the string in this picture are also fields Xm in a two-dimension field theory defined on the surface that a string sweeps out as it travels in space. The partial derivatives are with respect to the coordinates s and t on the world sheet and hmn is the two-dimensional metric defined on the string world sheet.
The general solution to the relativistic string equations of motion looks very similar to the classical nonrelativistic case above. The transverse space coordinates can be expanded in normal modes as

The string solution above is unlike a guitar string in that it isn't tied down at either end and so travels freely through spacetime as it oscillates. The string above is an open string, with ends that are floppy.
For a closed string, the boundary conditions are periodic, and the resulting oscillating solution looks like two open string oscillations moving in the opposite direction around the string. These two types of closed string modes are called right-movers and left-movers, and this difference will be important later in the supersymmetric heterotic string theory.
This is classical string. When we add quantum mechanics by making the string momentum and position obey quantum commutation relations, the oscillator mode coefficients have the commutation relations

The quantized string oscillator modes wind up giving representations of the Poincaré group, through which quantum states of mass and spin are classified in a relativistic quantum field theory.
So this is where the elementary particle arise in string theory. Particles in a string theory are like the harmonic notes played on a string with a fixed tension

String tension

The parameter a' is called the string parameter and the square root of this number represents the approximate distance scale at which string effects should become observable.
In the generic quantum string theory, there are quantum states with negative norm, also known as ghosts. This happens because of the minus sign in the spacetime metric, which implies that

So there ends up being extra unphysical states in the string spectrum.
In 26 spacetime dimensions, these extra unphysical states wind up disappearing from the spectrum. Therefore. bosonic string quantum mechanics is only consistent if the dimension of spacetime is 26.
By looking at the quantum mechanics of the relativistic string normal modes, one can deduce that the quantum modes of the string look just like the particles we see in spacetime, with mass that depends on the spin according to the formula

Regge formula

Remember that boundary conditions are important for string behavior. Strings can be open, with ends that travel at the speed of light, or closed, with their ends joined in a ring.
One of the particle states of a closed string has zero mass and two units of spin, the same mass and spin as a graviton, the particle that is supposed to be the carrier of the gravitational force.

DO IT!!!!!!!!!!!!!!!!!!!!!

The multidimensional continua that Riemann was concerned with are essentially instances of what is now known as a real n-dimensional smooth manifold. For brevity, call them ‘n-manifolds’. (For example, a sphere, a Möbius strip and the surface of a Henry Moore sculpture may be regarded as 2-manifolds; the spacetimes models of current cosmology are 4-manifolds.) Conditions (a) -- © provide a full characterization of n-manifolds.

(a) An n-manifold M is a set of points that can be pieced together from partially overlapping patches, such that every point of M lies in at least one patch.

(b) M is endowed with a neighborhood structure (a topology) such that, if U is a patch of M, there is a continuous one-one mapping f of U onto some region of Rn, with continuous inverse f-1. (Rn denotes here the collection of all real number n-tuples, with the standard topology generated by the open balls.) f is a coordinate system or chart of M; the k-th number in the n-tuple assigned by a chart f to a point P in f's patch is called the k-th coordinate of P by f; the k-th coordinate function of chart f is the real-valued function that assigns to each point of the patch its k-th coordinate by f.

© There is a collection A of charts of M, which contains at least one chart defined on each patch of M and is such that, if g and h belong to A, the composite mappings h g-1 and g h-1 --- known as coordinate transformations --- are differentiable to every order wherever they are well defined. (Denote the real number n-tuple <a1, ... , an> by a. The mapping h g-1 is well defined at a if a is the valued assigned by g to some point P of M to which h also assigns a value. Suppose that the latter value h(P) = <b1, ... ,bn> = b; then, b = h g-1(a). Since h g-1 maps a region of Rn into Rn, it makes sense to say that h g-1 is differentiable.) Such a collection A is called an atlas.[*]

It is the pair <M, A> that, strictly speaking, is an n-manifold, in the sense defined above. If <M1,A1> and <M2,A2> are an n-manifold and an m-manifold, respectively, it makes good sense to say that a mapping f of M1 into M2 is differentiable at a point P of M1 if, for a chart h defined at P and a chart g defined at f(P), the composite mapping g f h-1 is differentiable at h(P). (Condition © implies that the fulfillment of this requirement does not depend on the choice of h and g.) f is differentiable if it is differentiable at every point of M1.

Let <M,A> be an n-manifold. To each point P of M one associates a vector space, which is known as the tangent space at P and is denoted by TPM. The idea is based on the intuitive notion of a plane tangent to a surface at a given point. It can be constructed as follows. Let be a one-one differentiable mapping of a real open interval I into M. We can think of the successive values of as forming the path of a point that moves through M during a time interval represented by I. We call a curve in M (parametrized by u I). Put (t0) = P for a fixed number t0 in I. Consider the collection F(P) of all differentiable real-valued functions defined on some neighborhood of P. With the ordinary operations of function addition and multiplication by a constant, F(P) has the structure of a vector space. Each function f in F(P) varies smoothly with u, along the path of , in some neighborhood of P. Its rate of variation at P = (t0) is properly expressed by the derivative d(f)/du at u = t0. As f ranges over F(P), the value of d(f)/du at u = t0 is apt to vary in R. So we have here a mapping of F(P) into R, which we denote by (u). It is in fact a linear function and therefore a vector in the dual space F*(P) of real-valued linear functions on F(P). Call it the tangent to at P. The tangents at P to all the curves whose paths go through P span an n-dimensional subspace of F*(P). This subspace is, by definition, the tangent space TPM. The tangent spaces at all points of an n-manifold M can be bundled together in a natural way into a 2n-manifold TM. The projection mapping of TM onto M assigns to each tangent vector v in TPM the point (v) at which v is tangent to M. The structure <TM,M,> is the tangent bundle over M. A vector field on M is a section of TM, i.e., a differentiable mapping f of M into TM such that f sends each point P of M to itself; such a mapping obviously assigns to P a vector in TPM.

Any vector space V is automatically associated with other vector spaces, such as the dual space V* of linear functions on V, and the diverse spaces of multilinear functions on V, on V*, and on any possible combination of V and V*. This holds, of course, for each tangent space of an n-manifold M. The dual of TPM is known as the cotangent space at P. There is a natural way of bundling together the cotangent spaces of M into a 2n-manifold, the cotangent bundle. Generally speaking, all the vector spaces of a definite type associated with the tangent and cotangent spaces of M can be naturally bundled together into a k-manifolds (for suitable integers k, depending on the nature of the bundled items). A section of any of these bundles is a tensor field on M (of rank r, if the bundled objects are r-linear functions).

A Riemannian metric g on the n-manifold <M,A> is a tensor field of rank 2 on M. Thus, g assigns to each P in M a bilinear function gP on TPM. For any P in M and any vectors v, w, in TPM, gP must meet these requirements:

(i) gP(v,w) = gP(w,v) (symmetry)

(ii) gP(v,w) = 0 for all vectors w in TPM if and only if v is the 0-vector (non-degeneracy)

(iii) gP(v,v) > 0 unless v is the 0-vector (positive definiteness).

It is worth noting that the so-called Lorentzian metrics defined by relativity theory on its spacetime models meet requirements (i) and (ii), but not (iii), and are therefore usually said to be semi-Riemannian.

The length (v) of a vector v in TPM is defined by |(v)|2 = gP(v,v). Let be a curve in M. Let (u) be the tangent to at the point (u). The length of 's path from (a) to (b) is measured by the integral

((u))du

Thus, in Riemannian geometry, the length of the tangent vector (u) bears witness to the advance of curve g as it passes through the point (u). The definition of the length of a curve leads at once to the notion of a geodesic (or straightest) curve, which is characterized by the fact that its length is extremal; in other words, a geodesic is either the greatest or the shortest among all the curves that trace out neighboring paths between the same two points.

In his study of curved surfaces, Gauss introduced a real-valued function, the Gaussian curvature, which measures a surface's local deviation from flatness in terms of the surface's intrinsic geometry. Riemann extended this concept of curvature to Riemannian n-manifolds. He observed that each geodesic through a point in such a manifold is fully determined by its tangent vector at that point. Consider a point P in a Riemannian n-manifold <M,A,g> and two linearly independent vectors v and w in TPM. The geodesics determined by all linear combinations of v and w form a 2-manifold about P, with a definite Gaussian curvature KP(v,w) at P. The real number KP(v,w) measures the curvature of M at P in the ‘surface direction’ (Riemann 1854, p. 145) fixed by v and w. Riemann (1861) thought up a global mapping, depending on the metric g, that yields the said values KP(v,w) on appropriate arguments P, v and w. Nowadays this object is conceived as a tensor field of rank 4, which assigns to each point P in a Riemannian n-manifold <M,A,g> a 4-linear function on the tangent space TPM. It is therefore known as the Riemann tensor. Given the above definition of KP(v,w) it is clear that, if n = 2, the Riemann tensor reduces to the Gaussian curvature function.

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Calculus and Analysis > Calculus > Differential Calculus v
MathWorld Contributors > Derwent v

Derivative

COMMENT On this Page EXPLORE THIS TOPIC IN the MathWorld ClassroomDOWNLOAD Mathematica Notebook

The derivative of a function represents an infinitesimal change in the function with respect to whatever parameters it may have. The "simple" derivative of a function f with respect to a variable x is denoted either f^'(x) or
(df)/(dx), (1)

often written in-line as df/dx. When derivatives are taken with respect to time, they are often denoted using Newton's Eric Weisstein's World of Biography overdot notation for fluxions,
(dx)/(dt)==x^.. (2)

When a derivative is taken n times, the notation f^((n))(x) or
(d^nf)/(dx^n) (3)

is used, with
x^.,x^..,x^..., (4)

etc., the corresponding fluxion notation. When a function f(x,y,...) depends on more than one variable, a partial derivative
(partialf)/(partialx),(partial^2f)/(partialxpartialy), etc. (5)

can be used to specify the derivative with respect to one or more variables.

The derivative of a function f(x) with respect to the variable x is defined as
f^'(x)=lim_(h->0)(f(x+h)-f(x))/h, (6)

but may also be calculated more symmetrically as
f^'(x)==lim_(h->0)(f(x+h)-f(x-h))/(2h), (7)

provided the derivative is known to exist.

It should be noted that the above definitions refer to "real" derivatives, i.e., derivatives which are restricted to directions along the real axis. However, this restriction is artificial, and derivatives are most naturally defined in the complex plane, where they are sometimes explicitly referred to as complex derivatives. In order for complex derivatives to exist, the same result must be obtained for derivatives taken in any direction in the complex plane. Somewhat surprisingly, almost all of the important functions in mathematics satisfy this property, which is equivalent to saying that they satisfy the Cauchy-Riemann equations.

These considerations can lead to confusion for students because elementary calculus texts commonly consider only "real" derivatives, never alluding the the existence of complex derivatives, variables, or functions. For example, textbook examples to the contrary, the "derivative" (read: complex derivative) d|z|/dz of the absolute value function |z| does not exist because at every point in the complex plane, the value of the derivative depends on the direction in which the derivative is taken (so the Cauchy-Riemann equations cannot and do not hold). However, the real derivative (i.e., restricting the derivative to directions along the real axis) can be defined for points other than x==0 as
(d|x|)/(dx)=={-1 for x<0; undefined for x==0; 1 for x>0. (8)

As a result of the fact that computer algebra programs such as Mathematica generically deal with complex variables (i.e., the definition of derivative always means complex derivative), d|x|/dx correctly returns unevaluated by such software.

If the first derivative exists, the second derivative may be defined as
f^('')(x)=lim_(h->0)(f^'(x+h)-f^'(x))/h (9)

and calculated more symmetrically as
f^('')(x)==lim_(h->0)(f(x+2h)-2f(x+h)+f(x))/(h^2), (10)

again provided the second derivative is known to exist.

Note that in order for the limit to exist, both lim_(h->0^+) and lim_(h->0^-) must exist and be equal, so the function must be continuous. However, continuity is a necessary but not sufficient condition for differentiability. Since some discontinuous functions can be integrated, in a sense there are "more" functions which can be integrated than differentiated. In a letter to Stieltjes, Hermite Eric Weisstein's World of Biography wrote, "I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives."

A three-dimensional generalization of the derivative to an arbitrary direction is known as the directional derivative. In general, derivatives are mathematical objects which exist between smooth functions on manifolds. In this formalism, derivatives are usually assembled into "tangent maps."

Performing numerical differentiation is in many ways more difficult than numerical integration. This is because while numerical integration requires only good continuity properties of the function being integrated, numerical differentiation requires more complicated properties such as Lipschitz classes.

Simple derivatives of some simple functions follow.
d/(dx)x^n = nx^(n-1) (11)
d/(dx)lnx = 1/x (12)
d/(dx)sinx = cosx (13)
d/(dx)cosx = -sinx (14)
d/(dx)tanx = sec^2x (15)
d/(dx)cscx = -cscxcotx (16)
d/(dx)secx = secxtanx (17)
d/(dx)cotx = -csc^2x (18)
d/(dx)e^x = e^x (19)
d/(dx)a^x = (lna)a^x (20)
d/(dx)sin^(-1)x = 1/(sqrt(1-x^2)) (21)
d/(dx)cos^(-1)x = -1/(sqrt(1-x^2)) (22)
d/(dx)tan^(-1)x = 1/(1+x^2) (23)
d/(dx)cot^(-1)x = -1/(1+x^2) (24)
d/(dx)sec^(-1)x = 1/(xsqrt(x^2-1)) (25)
d/(dx)csc^(-1)x = -1/(xsqrt(x^2-1)) (26)
d/(dx)sinhx = coshx (27)
d/(dx)coshx = sinhx (28)
d/(dx)tanhx = sech^2x (29)
d/(dx)cothx = -csch^2x (30)
d/(dx)sechx = -sechxtanhx (31)
d/(dx)cschx = -cschxcothx (32)
d/(dx)snx = cnxdnx (33)
d/(dx)cnx = -snxdnx (34)
d/(dx)dnx = -k^2snxcnx. (35)

where sn(x)=sn(x,k), cn(x)=cn(x,k), etc. are Jacobi elliptic functions, and the product rule and quotient rule have been used extensively to expand the derivatives.

There are a number of important rules for computing derivatives of certain combinations of functions. Derivatives of sums are equal to the sum of derivatives so that
[f(x)+...+h(x)]^'==f^'(x)+...+h^'(x). (36)

In addition, if c is a constant,
d/(dx)[cf(x)]==cf^'(x). (37)

The product rule for differentiation states
d/(dx)[f(x)g(x)]==f(x)g^'(x)+f^'(x)g(x), (38)

where f^' denotes the derivative of f with respect to x. This derivative rule can be applied iteratively to yield derivative rules for products of three or more functions, for example,
[fgh]^' = (fg)h^'+(fg)^'h==fgh^'+(fg^'+f^'g)h (39)
= f^'gh+fg^'h+fgh^'. (40)

The quotient rule for derivatives states that
d/(dx)[(f(x))/(g(x))]==(g(x)f^'(x)-f(x)g^'(x))/([g(x)]^2) (41)

while the power rule gives
d/(dx)(x^n)==nx^(n-1). (42)

Other very important rule for computing derivatives is the chain rule, which states that for y==y(u),
(dy)/(dx)==(dy)/(du).(du)/(dx), (43)

or more generally, for z==z(x(t),y(t))
(dz)/(dt)==(partialz)/(partialx)(dx)/(dt)+(partialz)/(partialy)(dy)/(dt), (44)

where partialz/partialx denotes a partial derivative.

Miscellaneous other derivative identities include
(dy)/(dx)==((dy)/(dt))/((dx)/(dt)) (45)
(dy)/(dx)==1/((dx)/(dy)). (46)

If F(x,y)==C, where C is a constant, then
dF==(partialF)/(partialy)dy+(partialF)/(partialx)dx==0, (47)

so
(dy)/(dx)==-((partialF)/(partialx))/((partialF)/(partialy)). (48)

Derivative identities of inverse functions include
(dx)/(dy) = 1/((dy)/(dx)) (49)
(d^2x)/(dy^2) = -(d^2y)/(dx^2)((dy)/(dx))^(-3) (50)
(d^3x)/(dy^3) = [3((d^2y)/(dx^2))^2-(d^3y)/(dx^3)(dy)/(dx)]((dy)/(dx))^(-5). (51)

A vector derivative of a vector function
X(t)=[x_1(t); x_2(t); |; x_k(t)] (52)

can be defined by
(dX)/(dt)==[(dx_1)/(dt); (dx_2)/(dt); |; (dx_k)/(dt)]. (53)

The nth derivatives of x^nf(x) for n==1, 2, ... are
d/(dx)[xf(x)] = f(x)+xf^'(x) (54)
(d^2)/(dx^2)[x^2f(x)] = 2f(x)+4xf^'(x)+x^2f^('')(x) (55)
(d^3)/(dx^3)[x^3f(x)] = 6f(x)+18xf^'(x)+9x^2f^('')(x)+x^3f^(''')(x). (56)

The nth row of the triangle of coefficients 1; 1, 1; 2, 4, 1; 6, 18, 9, 1; ... (Sloane's A021009) is given by the absolute values of the coefficients of the Laguerre polynomial L_n(x).

Faá di Bruno's formula gives an explicit formula for the nth derivative of the composition f(g(t)).

SEE ALSO: Blancmange Function, Calculus, Carathéodory Derivative, Cauchy-Riemann Equations, Chain Rule, Comma Derivative, Complex Derivative, Complex Differentiable, Convective Derivative, Covariant Derivative, Definite Integral, Differentiable, Differential Calculus, Differentiation, Directional Derivative, Euler-Lagrange Derivative, Faá di Bruno's Formula, Finite Difference, Fluxion, Fractional Calculus, Fréchet Derivative, Functional Derivative, Implicit Differentiation, Indefinite Integral, Integral, Lagrangian Derivative, Lie Derivative, Logarithmic Derivative, Numerical Differentiation, Pincherle Derivative, Power Rule, Product Rule, q-Derivative, Quotient Rule, Schwarzian Derivative, Semicolon Derivative, Total Derivative, Weierstrass Function. [Pages Linking Here]


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