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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 (mathematics) is the greatest rate of change.
A generalization of the gradient, for functions on a Banach space which have vectorial values, is the Jacobian.
Interpretations of the gradient
Consider a room in which the temperature is given by a scalar field \phi, so at each point (x,y,z) the temperature is \phi(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#Slope of a road or railroad or
Grade (slope) at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector.
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°.
This observation can be mathematically stated as follows. The gradient of the hill height function H dot product with a unit
vector (spatial) gives the slope of the hill in the direction of the vector. This is called the directional derivative.
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.
By definition, the gradient is a
vector field whose components are the
partial derivative of f. That is:
\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)_x\cdot 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 matrixs, as it should be, in view of the geometric interpretation given above.
Because the gradient is orthogonal to level sets, it can be used to construct a vector normal to a surface. Consider any manifold that is one dimension less that the space it is in (i.e. a surface in 3D, a curve in 2D, etc.). Let this manifold be defined by an equation e.g.
F(
x,
y,
z) = 0 (i.e. move everything to one side of the equation). We have now turned the manifold into a level set. To find a normal vector, we simply need to find the gradient of the function
F at the desired point.
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.
Expressions for the gradient in 3 dimensions
The form of the gradient depends on the coordinate system used.
In
Cartesian coordinates, the above expression expands to
\nabla f(x, y, z) = \begin{pmatrix}{\frac{\partial f}{\partial x-->,{\frac{\partial f}{\partial y-->,{\frac{\partial f}{\partial z-->\end{pmatrix}.
In cylindrical coordinates,
\nabla f(\rho, \theta, z) = \begin{pmatrix}{\frac{\partial f}{\partial \rho-->,{\frac{1}{\rho}\frac{\partial f}{\partial \theta-->,{\frac{\partial f}{\partial z-->\end{pmatrix}
(where \theta is the azimuthal angle and z is the axial coordinate).
In spherical coordinates:
\nabla f(r, \theta, \phi) = \begin{pmatrix}{\frac{\partial f}{\partial r-->,{\frac{1}{r}\frac{\partial f}{\partial \theta-->,{\frac{1}{r \sin\theta}\frac{\partial f}{\partial \phi-->\end{pmatrix}
(where \theta is the azimuthal angle and \phi is the polar angle).
Example
For example, the gradient of the function in Cartesian coordinates
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 (mathematics) f from the Euclidean space \mathbb{R}^n to \mathbb{R} at any particular point
x0 in \mathbb{R}^n characterizes the best
linear approximation to
f at
x0. The approximation is as follows:
f(x) \approx f(x_0) + (\nabla f)_{x_0}\cdot(x-x_0)
for x close to x_0, where (\nabla f)_{x_0} is the gradient of
f computed at x_0, and the dot denotes the
dot product on \mathbb{R}^n. This equation is equivalent to the first two terms in the multi-variable Taylor Series expansion of
f at
x0.
The differential or (exterior) derivative
The best linear approximation to a function f : \mathbb{R}^n \to \mathbb{R} at a point x in \mathbb{R}^n is a linear map from \mathbb{R}^n to \mathbb{R} which is often denoted by \mathrm{d}f_x or Df(x) and called the differential (calculus) or
total derivative of f at x. The gradient is therefore related to the differential by the formula
(\nabla f)_x\cdot v = \mathrm d f_x(v)
for any v \in \mathbb{R}^n. The function \mathrm{d}f, which maps x to \mathrm{d}f_x, is called the differential or
exterior derivative of f and is an example of a differential 1-form.
If \mathbb{R}^n is viewed as the space of (length n) column vectors (of real numbers), then one can regard \mathrm{d}f as the row vector
\mathrm{d}f = \left( \frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n}\right)
so that \mathrm{d}f_x(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 covariance and contravariance of 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.
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.
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,
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
x∈
M 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:
\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 d
f using the
musical isomorphism \sharp=\sharp^g\colon T^*M\to 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.
See also
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References
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 (mathematics) is the greatest rate of change.
A generalization of the gradient, for functions on a
Banach space which have vectorial values, is the Jacobian.
Interpretations of the gradient
Consider a room in which the temperature is given by a scalar field \phi, so at each point (x,y,z) the temperature is \phi(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#Slope of a road or railroad or Grade (slope) at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector.
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°.
This observation can be mathematically stated as follows. The gradient of the hill height function H dot product with a unit vector (spatial) gives the slope of the hill in the direction of the vector. This is called the directional derivative.
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.
By definition, the gradient is a vector field whose components are the
partial derivative of f. That is:
\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)_x\cdot 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 matrixs, as it should be, in view of the geometric interpretation given above.
Because the gradient is orthogonal to level sets, it can be used to construct a vector normal to a surface. Consider any manifold that is one dimension less that the space it is in (i.e. a surface in 3D, a curve in 2D, etc.). Let this manifold be defined by an equation e.g.
F(
x,
y,
z) = 0 (i.e. move everything to one side of the equation). We have now turned the manifold into a level set. To find a normal vector, we simply need to find the gradient of the function
F at the desired point.
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.
Expressions for the gradient in 3 dimensions
The form of the gradient depends on the coordinate system used.
In
Cartesian coordinates, the above expression expands to
\nabla f(x, y, z) = \begin{pmatrix}{\frac{\partial f}{\partial x-->,{\frac{\partial f}{\partial y-->,{\frac{\partial f}{\partial z-->\end{pmatrix}.
In cylindrical coordinates,
\nabla f(\rho, \theta, z) = \begin{pmatrix}{\frac{\partial f}{\partial \rho-->,{\frac{1}{\rho}\frac{\partial f}{\partial \theta-->,{\frac{\partial f}{\partial z-->\end{pmatrix}
(where \theta is the azimuthal angle and z is the axial coordinate).
In
spherical coordinates:
\nabla f(r, \theta, \phi) = \begin{pmatrix}{\frac{\partial f}{\partial r-->,{\frac{1}{r}\frac{\partial f}{\partial \theta-->,{\frac{1}{r \sin\theta}\frac{\partial f}{\partial \phi-->\end{pmatrix}
(where \theta is the azimuthal angle and \phi is the polar angle).
Example
For example, the gradient of the function in Cartesian coordinates
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 (mathematics) f from the
Euclidean space \mathbb{R}^n to \mathbb{R} at any particular point
x0 in \mathbb{R}^n characterizes the best
linear approximation to
f at
x0. The approximation is as follows:
f(x) \approx f(x_0) + (\nabla f)_{x_0}\cdot(x-x_0)
for x close to x_0, where (\nabla f)_{x_0} is the gradient of
f computed at x_0, and the dot denotes the
dot product on \mathbb{R}^n. This equation is equivalent to the first two terms in the multi-variable Taylor Series expansion of
f at
x0.
The differential or (exterior) derivative
The best linear approximation to a function f : \mathbb{R}^n \to \mathbb{R} at a point x in \mathbb{R}^n is a linear map from \mathbb{R}^n to \mathbb{R} which is often denoted by \mathrm{d}f_x or Df(x) and called the
differential (calculus) or
total derivative of f at x. The gradient is therefore related to the differential by the formula
(\nabla f)_x\cdot v = \mathrm d f_x(v)
for any v \in \mathbb{R}^n. The function \mathrm{d}f, which maps x to \mathrm{d}f_x, is called the differential or
exterior derivative of f and is an example of a
differential 1-form.
If \mathbb{R}^n is viewed as the space of (length n) column vectors (of real numbers), then one can regard \mathrm{d}f as the row vector
\mathrm{d}f = \left( \frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n}\right)
so that \mathrm{d}f_x(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 covariance and contravariance of 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.
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.
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,
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
x∈
M 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:
\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 d
f using the musical isomorphism \sharp=\sharp^g\colon T^*M\to 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.
See also
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| width=40 ||
|}
References
NEWS - Gradient Wings
Gradient Wings - Home of Gradient Paragliders in the UK ... The BPCup winner for 2008 is Cris Miles flying his Avax XC2! This year was particularly hard on the pilots as the ...
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Finnish manufacturer of floor standing loudspeaker and subwoofer.
Gradient Index Prototype - Homepage
Gradient Index Prototype Tool. Welcome to Gradient Index Gradient: promoting best practice management of supply chain labour standards
Definition: gradient from Online Medical Dictionary
The Online Medical Dictionary is a searchable dictionary of definitions from medicine, science and technology.
Gradient - Wikipedia, the free encyclopedia
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 ...
Gradient -- from Wolfram MathWorld
The gradient is a vector operator denoted del and sometimes also called Del or nabla. It is most often applied to a real function of three variables f(u_1,u_2,u_3), and may be ...
Gradient Consulting
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25. 06. 2008 - NEW Glider MONTANA Montana is Gradient's new light weight mountain glider. With Montana, Gradient follows the success of our well known Delite glider which has been ...
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A selection of Gradient buttons for you to use on your own site. Designed in many styles and colours
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