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Gradient vs. Jacobian

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Gradientnoun

A slope or incline.

Jacobianadjective

alternative case form of Jacobian

Gradientnoun

A rate of inclination or declination of a slope.

Jacobiannoun

alternative case form of Jacobian

Gradientnoun

(calculus) Of a function y = f(x) or the graph of such a function, the rate of change of y with respect to x
that is, the amount by which y changes for a certain (often unit) change in x
equivalently, the inclination to the X axis of the tangent to the curve of the graph.

Jacobianadjective

relating to the work of the mathematician K. G. J. Jacobi.

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Gradientnoun

(science) The rate at which a physical quantity increases or decreases relative to change in a given variable, especially distance.

Jacobiannoun

a determinant whose constituents are the derivatives of a number of functions (u, v, w, …) with respect to each of the same number of variables (x, y, z, …).

Gradientnoun

(analysis) A differential operator that maps each point of a scalar field to a vector pointed in the direction of the greatest rate of change of the scalar. Notation for a scalar field φ: ∇φ

Gradientnoun

A gradual change in color. A color gradient; gradation.

Gradientadjective

Moving by steps; walking.

‘gradient automata’;

Gradientadjective

Rising or descending by regular degrees of inclination.

‘the gradient line of a railroad’;

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Gradientadjective

Adapted for walking, as the feet of certain birds.

Gradientadjective

Moving by steps; walking; as, gradient automata.

Gradientadjective

Rising or descending by regular degrees of inclination; as, the gradient line of a railroad.

Gradientadjective

Adapted for walking, as the feet of certain birds.

Gradientnoun

The rate of regular or graded ascent or descent in a road; grade.

Gradientnoun

A part of a road which slopes upward or downward; a portion of a way not level; a grade.

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Gradientnoun

The rate of increase or decrease of a variable magnitude, or the curve which represents it; as, a thermometric gradient.

Gradientnoun

The variation of the concentration of a chemical substance in solution through some linear path; also called concentration gradient; - usually measured in concentration units per unit distance. Concentration gradients are created naturally, e.g. by the diffusion of a substance from a point of high concentration toward regions of lower concentration within a body of liquid; in laboratory techniques they may be made artificially.

Gradientnoun

a graded change in the magnitude of some physical quantity or dimension

Gradientnoun

the property possessed by a line or surface that departs from the horizontal;

‘a five-degree gradient’;

Gradient

In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) ∇ f {\displaystyle \nabla f} whose value at a point p {\displaystyle p} is the vector whose components are the partial derivatives of f {\displaystyle f} at p {\displaystyle p} . That is, for f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } , its gradient ∇ f : R n → R n {\displaystyle \nabla f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} is defined at the point p = ( x 1 , … , x n ) {\displaystyle p=(x_{1},\ldots ,x_{n})} in n-dimensional space as the vector: ∇ f ( p ) = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] .

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