Gradient

In mathematics, the gradient is a generalization of the usual concept of derivative of a function in one dimension to a function in several dimensions. If f(x1, ..., xn) is a differentiable, scalar-valued function of standard Cartesian coordinates in Euclidean space, its gradient is the vector whose components are the n partial derivatives of f. It is thus a vector-valued function.

Similarly to the usual derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. This characterizing property of the gradient allows it to be defined independently of a choice of coordinate system, as a vector field whose components in a coordinate system will transform when going from one coordinate system to another.

Gradient (disambiguation)

The Gradient is the rate of variation of a numerical quantity. It may refer to:

  • Slope or grade, referring to the inclination of a road or other geographic features
  • Image gradient, a gradual change or blending of color
  • Color gradient
  • Texture gradient
  • Mathematics

  • Gradient, in vector calculus, a vector field representing the maximum rate of increase of a scalar field or a multivariate function and the direction of this maximal rate.
  • Gradient descent
  • Gradient theorem
  • Gradient method
  • Conjugate gradient method
  • Nonlinear conjugate gradient method
  • Stochastic gradient descent
  • Biology

  • Concentration gradient, the ratio of solute concentration between two adjoining regions
  • Potential gradient, the difference in electric charge between two adjoining regions
  • Fluid dynamics and earth science

  • Density gradient
  • Pressure gradient
  • Temperature gradient
  • Geothermal gradient
  • Sound speed gradient
  • Wind gradient
  • See also

    Four-gradient

    In differential geometry, the four-gradient is the four-vector analogue of the gradient from Gibbs-Heaviside vector calculus.

    Definition

    The covariant components compactly written in index notation are:

    The comma in the last part above  {}_{,\alpha} implies the partial differentiation with respect to x^\alpha. This is not the same as a semi-colon, used for the covariant derivative.

    The contravariant components are:

    where gαβ is the metric tensor, which here has been chosen for flat spacetime with the metric signature (+,−,−,−).

    Alternative symbols to \partial_\alpha are \Box and D.

    Usage

    The 4-Gradient is used in a number of different ways in Special Relativity:


    As a 4-Divergence

    The 4-Divergence of the 4-Position X^\mu gives the dimensionality of spacetime:


    The 4-Divergence of the 4-CurrentDensity J^\mu = \rho_o U^\mu gives a conservation law - the conservation of charge:

    This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density \partial_t \rho = -\vec{\nabla}\cdot \vec{j}. In other words, the charge inside a box cannot just change arbitrarily, it must enter and leave the box via a current. This is a continuity equation.

    Podcasts:

    PLAYLIST TIME:

    In Anger

    by: Cardiant

    Why I thought I had it all
    and why I wanted more
    Could it be just there for while,
    the while I need for myself
    Never dared to seal my door
    Never dared to look behind
    Once again I was
    As fool as I can be
    Anger takes my mind at will
    The will I never could lead
    Feel this anxious rise
    Got no chance to be myself
    Maybe somewhere I´ll find
    the place where I can just be
    Let me allow this dawn
    Before we´ll follow that star
    I´m making my command
    on this hazy unknown ground
    (The) Pain and the anger is gone
    I know my life is worth of this run
    I won´t again put myself to burn my
    Candle at both ends
    Once we will break our time
    And that shine in your eyes will be my guide
    Looking into the stars
    made me blindly believe
    in the truth of the lighting star
    It may be hard to tell in advance




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