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Encyclopedia > Gradient
In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.
In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change. Look up gradient in Wiktionary, the free dictionary. ... Image File history File links Gradient2. ... Image File history File links Gradient2. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ...


A generalization of the gradient, for functions which have vectorial values, is the Jacobian. In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

Contents

Interpretations of the gradient

Consider a room in which the temperature is given by a scalar field φ, so at each point (x,y,z) the temperature is φ(x,y,z). We will assume that the temperature does not change in time. Then, at each point in the room, the gradient at that point will show the direction in which the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.


Consider a hill whose height above sea level at a point (x,y) is H(x,y). The gradient of H at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector. Look up Slope in Wiktionary, the free dictionary. ... A grade (or gradient) is the pitch of a slope, and is often expressed as a percent tangent, or rise over run. It is used to express the steepness of slope on a hill, roof, or road, where zero indicates level (with respect to gravity) and increasing numbers correlate to...


The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a dot product. Consider again the example with the hill and suppose that the steepest slope on the hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%. If instead, the road goes around the hill at an angle with the uphill direction (the gradient vector), then it will have a shallower slope. For example, if the angle between the road and the uphill direction, projected onto the horizontal plane, is 60°, then the steepest slope along the road will be 20% which is 40% times the cosine of 60°. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...


This observation can be mathematically stated as follows. The gradient of the hill height function H dotted with a unit vector gives the slope of the hill in the direction of the vector. This is called the directional derivative. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the...


Formal definition

The gradient (or gradient vector field) of a scalar function f(x) with respect to a vector variable x = (x_1,dots,x_n) is denoted by nabla f or vec{nabla} f where nabla (the nabla symbol) denotes the vector differential operator del. The notation operatorname{grad}(f) is also used for the gradient. Nabla is a symbol, shown as . ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In vector calculus, del is a vector differential operator represented by the nabla symbol: ∇. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ...


By definition, the gradient is a vector field whose components are the partial derivatives of f. That is: Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...

nabla f = left(frac{partial f}{partial x_1 }, dots, frac{partial f}{partial x_n } right)

(Here the gradient is written as a row vector, but it is often taken to be a column vector.)


The dot product (nabla f)_xcdot v of the gradient at a point x with a vector v gives the directional derivative of f at x in the direction v. It follows that the gradient of f is orthogonal to the level sets of f. This also shows that, although the gradient was defined in terms of coordinates, it is actually invariant under orthogonal transformations, as it should be, in view of the geometric interpretation given above. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...


The gradient is an irrotational vector field and line integrals through a gradient field are path independent and can be evaluated with the gradient theorem. Conversely, an irrotational vector field in a simply connected region is always the gradient of a function. In vector calculus, an irrotational or conservative vector field is a vector field whose curl is zero. ... Rombu is the hawt. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...


Example

In 3 dimensions, the expression expands to nabla f = begin{pmatrix} {frac{partial f}{partial x}}, {frac{partial f}{partial y}}, {frac{partial f}{partial z}} end{pmatrix} in Cartesian coordinates. For example, the gradient of the function :For other senses of this word, see dimension (disambiguation). ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

f(x,y,z)=  2x+3y^2-sin(z)

is:

nabla f= begin{pmatrix} {frac{partial f}{partial x}}, {frac{partial f}{partial y}}, {frac{partial f}{partial z}} end{pmatrix} = begin{pmatrix} {2}, {6y}, {-cos(z)} end{pmatrix}.

The gradient and the derivative or differential

Linear approximation to a function

The gradient of a function f from the Euclidean space Rn to R at any particular point x0 in Rn characterizes the best linear approximation to f at x0. The approximation is as follows: Partial plot of a function f. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Linear approximation is a method of approximating otherwise difficult to find values of a mathematical function by taking the value on a nearby tangent line instead of the function itself. ...

f(x) approx f(x_0) + (nabla f)_{x_0}cdot(x-x_0)

for x close to x0, where (nabla f)_{x_0} is the gradient of f computed at x0, and the dot denotes the dot product on Rn. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...


The differential or (exterior) derivative

The best linear approximation to a function f:RnR at a point x in Rn is a linear map from Rn to R which is often denoted by dfx or Df(x) and called the differential or (total) derivative of f at x. The gradient is therefore related to the differential by the formula The differential dy In calculus, a differential is an infinitesimally small change in a variable. ... In mathematics, a total derivative may be either. ...

(nabla f)_xcdot v = mathrm d f_x(v)

for any vRn. The function df, which maps x to dfx, is called the differential or exterior derivative of f and is an example of a differential 1-form. In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...


If Rn is viewed as the space of (length n) column vectors (of real numbers), then one can regard df as the row vector

mathrm d f =( frac{partial f}{partial x_1 }, dots, frac{partial f}{partial x_n } )

so that dfx(v) is given by matrix multiplication. The gradient is then the corresponding column vector, i.e., nabla f = mathrm d f^T.


The covariance of the gradient

The differential is more natural than the gradient because it is invariant under all coordinate transformations (or diffeomorphisms), whereas the gradient is only invariant under orthogonal transformations (because of the implicit use of the dot product in its definition). Because of this, it is common to blur the distinction between the two concepts using the notion of covariant and contravariant vectors. From this point of view, the components of the gradient transform covariantly under changes of coordinates, so it is called a covariant vector field, whereas the components of a vector field in the usual sense transform contravariantly. In this language the gradient is the differential, as a covariant vector field is the same thing as a differential 1-form[1]. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... It has been suggested that this article or section be merged into Covariant transformation. ...


The gradient on Riemannian manifolds

For any smooth function f on a Riemannian manifold (M,g), the gradient of f is the vector field nabla f such that for any vector field X, In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

g(nabla f, X ) = partial_X f, qquad text{i.e.,}quad g_x((nabla f)_x, X_x ) = (partial_X f) (x)

where g_x( cdot, cdot ) denotes the inner product of tangent vectors at x defined by the metric g and partial_X f (sometimes denoted X(f)) is the function that takes any point xM to the directional derivative of f in the direction X, evaluated at x. In other words, in a coordinate chart varphi from an open subset of M to an open subset of Rn, (partial_X f)(x) is given by: In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

sum_{j=1}^n X^{j} (varphi(x)) frac{partial}{partial x_{j}}(f circ varphi^{-1}) Big|_{varphi(x)},

where Xj denotes the jth component of X in this coordinate chart.


Generalizing the case M=Rn, the gradient of a function is related to its exterior derivative, since (partial_X f) (x) = df_x(X_x). More precisely, the gradient nabla f is the vector field associated to the differential 1-form df using the musical isomorphism sharp=sharp^gcolon T^*Mto TM (called "sharp") defined by the metric g. The relation between the exterior derivative and the gradient of a function on Rn is a special case of this in which the metric is the flat metric given by the dot product. In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the musical isomorphism is an isomorphism between the tangent bundle and the cotangent bundle of a Riemannian manifold given by its metric. ...


References

  1. ^ Unfortunately this confusing language is confused further by differing conventions. Although the components of a differential 1-form transform covariantly under coordinate transformations, differential 1-forms themselves transform contravariantly (by pullback) under diffeomorphism. For this reason differential 1-forms are sometimes said to be contravariant rather than covariant, in which case vector fields are covariant rather than contravariant!

See also

Look up Gradient in
Wiktionary, the free dictionary.

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