Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 246 views

    Directional Derivative

    Uploaded by

    Jose Luis Condori
    The directional derivative of a function represents the instantaneous rate of change of the function along a given vector or direction at a given point. It generalizes the notion of a partial derivative by allowing the rate of change to be calculated along any direction rather than just along the coordinate axes. The directional derivative is defined as the limit of the difference quotient as it approaches along the given vector. For differentiable functions, the directional derivative can also be expressed in terms of the gradient of the function and the given vector. Directional derivatives have properties similar to ordinary derivatives, such as sum, constant multiple, product and chain rules.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd

    Directional Derivative

    Uploaded by

    Jose Luis Condori
    246 views5 pages
    The directional derivative of a function represents the instantaneous rate of change of the function along a given vector or direction at a given point. It generalizes the notion of a partial derivative by allowing the rate of change to be calculated along any direction rather than just along the coordinate axes. The directional derivative is defined as the limit of the difference quotient as it approaches along the given vector. For differentiable functions, the directional derivative can also be expressed in terms of the gradient of the function and the given vector. Directional derivatives have properties similar to ordinary derivatives, such as sum, constant multiple, product and chain rules.

    Original Description:

    Copyright

    © Attribution Non-Commercial (BY-NC)

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    The directional derivative of a function represents the instantaneous rate of change of the function along a given vector or direction at a given point. It generalizes the notion of a partial derivative by allowing the rate of change to be calculated along any direction rather than just along the coordinate axes. The directional derivative is defined as the limit of the difference quotient as it approaches along the given vector. For differentiable functions, the directional derivative can also be expressed in terms of the gradient of the function and the given vector. Directional derivatives have properties similar to ordinary derivatives, such as sum, constant multiple, product and chain rules.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    246 views5 pages

    Directional Derivative

    Uploaded by

    Jose Luis Condori
    The directional derivative of a function represents the instantaneous rate of change of the function along a given vector or direction at a given point. It generalizes the notion of a partial derivative by allowing the rate of change to be calculated along any direction rather than just along the coordinate axes. The directional derivative is defined as the limit of the difference quotient as it approaches along the given vector. For differentiable functions, the directional derivative can also be expressed in terms of the gradient of the function and the given vector. Directional derivatives have properties similar to ordinary derivatives, such as sum, constant multiple, product and chain rules.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 5

    Directional derivative

    Directional derivative
    Calculus

    Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem

    In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gteaux derivative.

    Definition
    The directional derivative of a scalar function

    along a vector

    is the function defined by the limit

    If the function f is differentiable at x, then the directional derivative exists along any vector v, and one has on the right denotes the gradient and is the dot product.[1] At any point x, the directional derivative with respect to time when it is moving at a speed and direction

    where the

    of f intuitively represents the rate of change of given by v at the point x.

    Some authors define the directional derivative to be with respect to the vector v after normalization, thus ignoring its magnitude. In this case, one has

    or in case f is differentiable at x,

    This definition has some disadvantages: it applies only when the norm of a vector is defined and nonzero. It is incompatible with notation used in some other areas of mathematics, physics and engineering, but should be used when what is wanted is the rate of increase in f per unit distance.

    Directional derivative

    Notation
    Directional derivatives can be also denoted by:

    where v is a parameterization of a curve to which v is tangent and which determines its magnitude.

    Properties
    Many of the familiar properties of the ordinary derivative hold for the directional derivative. These include, for any functions f and g defined in a neighborhood of, and differentiable at, p: 1. The sum rule: 2. The constant factor rule: For any constant c, 3. The product rule (or Leibniz rule): 4. The chain rule: If g is differentiable at p and h is differentiable at g(p), then

    In differential geometry
    Let M be a differentiable manifold and p a point of M. Suppose that f is a function defined in a neighborhood of p, and differentiable at p. If v is a tangent vector to M at p, then the directional derivative of f along v, denoted variously as (see covariant derivative), (see Lie derivative), or (see Tangent space Definition via derivations), can be defined as follows. Let : [1,1] M be a differentiable curve with (0) = p and (0) = v. Then the directional derivative is defined by

    This definition can be proven independent of the choice of , provided is selected in the prescribed manner so that (0) = v.

    Normal derivative
    A normal derivative is a directional derivative taken in the direction normal (that is, orthogonal) to some surface in space, or more generally along a normal vector field orthogonal to some hypersurface. See for example Neumann boundary condition. If the normal direction is denoted by , then the directional derivative of a function f is sometimes denoted as . In other notations

    Directional derivative

    In the continuum mechanics of solids


    Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors.[2] The directional directive provides a systematic way of finding these derivatives. The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.

    Derivatives of scalar valued functions of vectors


    Let be a real valued function of the vector is defined as . Then the derivative of with respect to (or at ) in the direction

    for all vectors Properties: 1. If 2. If 3. If

    then then then

    Derivatives of vector valued functions of vectors


    Let be a vector valued function of the vector is the vector defined as . Then the derivative of with respect to (or at ) in the direction

    for all vectors Properties: 1. If 2. If 3. If

    then then then

    Directional derivative

    Derivatives of scalar valued functions of second-order tensors


    Let (or at be a real valued function of the second order tensor ) in the direction . Then the derivative of with respect to is the second order tensor defined as

    for all second order tensors Properties: 1. If 2. If 3. If

    then then then

    Derivatives of tensor valued functions of second-order tensors


    Let be a second order tensor valued function of the second order tensor (or at ) in the direction . Then the derivative of with respect to is the fourth order tensor defined as

    for all second order tensors Properties: 1. If 2. If 3. If 4. If

    then then then then

    Notes
    [1] Technically, the gradient f is a covector, and the "dot product" is the action of this covector on the vector v. [2] J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover.

    References
    Hildebrand, F. B. (1976). Advanced Calculus for Applications. Prentice Hall. ISBN0-13-011189-9. K.F. Riley, M.P. Hobson, S.J. Bence (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN978-0-521-86153-3.

    External links
    Directional derivatives (http://mathworld.wolfram.com/DirectionalDerivative.html) at MathWorld. Directional derivative (http://planetmath.org/directionalderivative) at PlanetMath.

    Article Sources and Contributors

    Article Sources and Contributors


    Directional derivative Source: http://en.wikipedia.org/w/index.php?oldid=561495237 Contributors: 2001:4170:2000:1:0:0:0:31, Amelio Vzquez, Angela, Aretakis, Bbanerje, BenFrantzDale, Betacommand, Charles Matthews, CsDix, DVdm, Dbroadwell, Denisarona, Dysprosia, EconoPhysicist, Ehaussecker, F=q(E+v^B), Fanyavizuri, Fgnievinski, Fintor, Fredrik, Gauge, Giftlite, Ht686rg90, Ilya Voyager, Je at uwo, Jekter, Jogers, Kepke, Kevin Ryde, Kjetil1001, Linas, LokiClock, Magioladitis, MarSch, Michael Hardy, Myasuda, NCurse, Netrapt, O18, Oleg Alexandrov, PV=nRT, Paquitotrek, Point-set topologist, Quondum, Rausch, Reedy, S.arseneau, Silly rabbit, Snigbrook, Ssd, TheObtuseAngleOfDoom, Tkuvho, Tobias Bergemann, User A1, Usuwiki, Volland, WISo, WhiteHatLurker, WriterHound, 54 anonymous edits

    License
    Creative Commons Attribution-Share Alike 3.0 //creativecommons.org/licenses/by-sa/3.0/

    You might also like