Gradients - Revision history

Khanh: /* Definition */

2022-01-20T22:05:47Z

Definition

Khanh at 21:59, 20 January 2022

2022-01-20T21:59:20Z

Khanh at 19:01, 14 November 2021

2021-11-14T19:01:30Z

Lila: /* Riemannian manifolds */

2021-10-19T15:57:43Z

Riemannian manifolds

Lila: /* Gradient of a vector field */

2021-10-19T15:56:51Z

Gradient of a vector field

Lila: /* Generalizations */

2021-10-19T15:55:14Z

Generalizations

Lila: /* Generalizations */

2021-10-19T15:53:53Z

Generalizations

Lila: /* Cylindrical and spherical coordinates */

2021-10-19T15:51:18Z

Cylindrical and spherical coordinates

Lila at 15:50, 19 October 2021

2021-10-19T15:50:50Z

Lila: /* Notation */

2021-10-19T15:49:32Z

Notation

@@ Line 27: / Line 27: @@
 ==Definition==
-[[File:3d-gradient-cos.svg|thumb|350px|The gradient of the function {{math|''f''(''x'',''y'') {{=}} −(cos<sup>2</sup>''x'' + cos<sup>2</sup>''y'')<sup>2</sup>}} depicted as a projected vector field on the bottom plane.]]
+[[File:3d-gradient-cos.svg|thumb|350px|The gradient of the function ''f''(''x'',''y'') = −(cos<sup>2</sup>''x'' + cos<sup>2</sup>''y'')<sup>2</sup> depicted as a projected vector field on the bottom plane.]]
-The gradient (or gradient vector field) of a scalar function {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, …, ''x<sub>n</sub>'')}} is denoted {{math|∇''f''}} or {{math|{{vec|∇}}''f''}}  where {{math|∇}} (nabla) denotes the vector differential operator, del. The notation {{math|grad ''f''}}  is also commonly used to represent the gradient. The gradient of {{math|''f''}} is defined as the unique vector field whose dot product with any vector {{math|'''v'''}} at each point {{math|''x''}} is the directional derivative of {{math|''f''}} along {{math|'''v'''}}. That is,
+The gradient (or gradient vector field) of a scalar function {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, …, ''x<sub>n</sub>'')}} is denoted <math> \nabla f </math> or <math>\vec \nabla f </math>  where {{math|∇}} (nabla) denotes the vector differential operator, del. The notation {{math|grad ''f''}}  is also commonly used to represent the gradient. The gradient of {{math|''f''}} is defined as the unique vector field whose dot product with any vector {{math|'''v'''}} at each point {{math|''x''}} is the directional derivative of {{math|''f''}} along {{math|'''v'''}}. That is,
 :<math>\big(\nabla f(x)\big)\cdot \mathbf{v} = D_{\mathbf v}f(x).</math>

@@ Line 27: / Line 27: @@
 ==Definition==
-[[File:3d-gradient-cos.svg|thumb|350px|The gradient of the function {{math|''f''(''x'',''y'') {{=}} −(cos<sup>2</sup>''x'' + cos<sup>2</sup>''y'')<sup>2</sup>}} depicted as a projected [[vector field]] on the bottom plane.]]
+[[File:3d-gradient-cos.svg|thumb|350px|The gradient of the function {{math|''f''(''x'',''y'') {{=}} −(cos<sup>2</sup>''x'' + cos<sup>2</sup>''y'')<sup>2</sup>}} depicted as a projected vector field on the bottom plane.]]
 The gradient (or gradient vector field) of a scalar function {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, …, ''x<sub>n</sub>'')}} is denoted {{math|∇''f''}} or {{math|{{vec|∇}}''f''}}  where {{math|∇}} (nabla) denotes the vector differential operator, del. The notation {{math|grad ''f''}}  is also commonly used to represent the gradient. The gradient of {{math|''f''}} is defined as the unique vector field whose dot product with any vector {{math|'''v'''}} at each point {{math|''x''}} is the directional derivative of {{math|''f''}} along {{math|'''v'''}}. That is,

@@ Line 143: / Line 143: @@
 (called "sharp") defined by the metric {{math|''g''}}. The relation between the exterior derivative and the gradient of a function on {{math|'''R'''<sup>''n''</sup>}} is a special case of this in which the metric is the flat metric given by the dot product.
-==Resources==
+== Licensing ==
-* [https://en.wikipedia.org/wiki/Gradient Gradient], Wikipedia
+Content obtained and/or adapted from:

@@ Line 131: / Line 131: @@
 where {{math|''g''<sub>''x''</sub>( , )}} denotes the inner product of tangent vectors at {{math|''x''}} defined by the metric {{math|''g''}} and {{math|∂<sub>''X''</sub>&thinsp;''f''}} is the function that takes any point {{math|''x'' ∈ ''M''}} to the directional derivative of {{math|''f''}} in the direction {{math|''X''}}, evaluated at {{math|''x''}}. In other words, in a coordinate chart {{math|''φ''}} from an open subset of {{math|''M''}} to an open subset of {{math|'''R'''<sup>''n''</sup>}}, {{math|(∂<sub>''X''</sub>&thinsp;''f''&thinsp;)(''x'')}} is given by:
 :<math>\sum_{j=1}^n X^{j} \big(\varphi(x)\big) \frac{\partial}{\partial x_{j}}(f \circ \varphi^{-1}) \Bigg|_{\varphi(x)},</math>
-where {{math|''X''{{isup|''j''}}}} denotes the {{math|''j''}}th component of {{math|''X''}} in this coordinate chart.
+where {{math|''X''<sup>''j''</sup>}} denotes the {{math|''j''}}th component of {{math|''X''}} in this coordinate chart.
 So, the local form of the gradient takes the form:

@@ Line 104: / Line 104: @@
 Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensor quantity.
-In rectangular coordinates, the gradient of a vector field {{math|1='''f''' = (&thinsp;''f''<sup>1</sup>, ''f''{{i sup|2}}, ''f''{{i sup|3}})}} is defined by:
+In rectangular coordinates, the gradient of a vector field {{math|1='''f''' = (&thinsp;''f''<sup>''1''</sup>, ''f''<sup>''2''</sup>, ''f''<sup>''3''</sup>)}} is defined by:
 :<math>\nabla \mathbf{f}=g^{jk}\frac{\partial f^i}{\partial x^j} \mathbf{e}_i \otimes \mathbf{e}_k,</math>
@@ Line 116: / Line 116: @@
 :<math>\nabla \mathbf{f}=g^{jk}\left(\frac{\partial f^i}{\partial x^j}+{\Gamma^i}_{jl}f^l\right) \mathbf{e}_i \otimes \mathbf{e}_k,</math>
-where {{math|''g''{{i sup|''jk''}}}} are the components of the inverse metric tensor and the {{math|'''e'''<sub>''i''</sub>}} are the coordinate basis vectors.
+where {{math|''g''<sup>''jk''</sup>}} are the components of the inverse metric tensor and the {{math|'''e'''<sub>''i''</sub>}} are the coordinate basis vectors.
 Expressed more invariantly, the gradient of a vector field {{math|'''f'''}} can be defined by the Levi-Civita connection and metric tensor:

@@ Line 83: / Line 83: @@
 === Jacobian ===
-The [[Jacobian matrix]] is the generalization of the gradient for vector-valued functions of several variables and [[differentiable map]]s between [[Euclidean space]]s or, more generally, [[manifold]]s.<ref>{{harvtxt|Beauregard|Fraleigh|1973|pp=87,248}}</ref><ref>{{harvtxt|Kreyszig|1972|pp=333,353,496}}</ref>  A further generalization for a function between [[Banach space]]s is the [[Fréchet derivative]].
+The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative.
 Suppose {{math|'''f''' : ℝ<sup>''n''</sup> → ℝ<sup>''m''</sup>}} is a function such that each of its first-order partial derivatives exist on {{math|ℝ<sup>''n''</sup>}}.  Then the Jacobian matrix of {{math|'''f'''}} is defined to be an {{math|''m''×''n''}} matrix, denoted by <math>\mathbf{J}_\mathbb{f}(\mathbb{x})</math> or simply <math>\mathbf{J}</math>. The {{math|(''i'',''j'')}}th entry is <math>\mathbf J_{ij} = \frac{\partial f_i}{\partial x_j}</math>. Explicitly
@@ Line 102: / Line 101: @@
 ===Gradient of a vector field===
-{{see also|Covariant derivative}}
-Since the total derivative of a vector field is a [[linear mapping]] from vectors to vectors, it is a [[tensor]] quantity.
+Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensor quantity.
 In rectangular coordinates, the gradient of a vector field {{math|1='''f''' = (&thinsp;''f''{{i sup|1}}, ''f''{{i sup|2}}, ''f''{{i sup|3}})}} is defined by:
@@ Line 109: / Line 108: @@
 :<math>\nabla \mathbf{f}=g^{jk}\frac{\partial f^i}{\partial x^j} \mathbf{e}_i \otimes \mathbf{e}_k,</math>
-(where the [[Einstein summation notation]] is used and the [[tensor product]] of the vectors {{math|'''e'''<sub>''i''</sub>}} and {{math|'''e'''<sub>''k''</sub>}} is a [[dyadic tensor]] of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix:
+(where the Einstein summation notation is used and the tensor product of the vectors {{math|'''e'''<sub>''i''</sub>}} and {{math|'''e'''<sub>''k''</sub>}} is a dyadic tensor of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix:
 :<math>\frac{\partial f^i}{\partial x^j} = \frac{\partial (f^1,f^2,f^3)}{\partial (x^1,x^2,x^3)}.</math>
-In curvilinear coordinates, or more generally on a curved [[Riemannian manifold|manifold]], the gradient involves [[Christoffel symbols]]:
+In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols:
 :<math>\nabla \mathbf{f}=g^{jk}\left(\frac{\partial f^i}{\partial x^j}+{\Gamma^i}_{jl}f^l\right) \mathbf{e}_i \otimes \mathbf{e}_k,</math>
-where {{math|''g''{{i sup|''jk''}}}} are the components of the inverse [[metric tensor]] and the {{math|'''e'''<sub>''i''</sub>}} are the coordinate basis vectors.
+where {{math|''g''{{i sup|''jk''}}}} are the components of the inverse metric tensor and the {{math|'''e'''<sub>''i''</sub>}} are the coordinate basis vectors.
-Expressed more invariantly, the gradient of a vector field {{math|'''f'''}} can be defined by the [[Levi-Civita connection]] and metric tensor:<ref>{{harvnb|Dubrovin|Fomenko|Novikov|1991|pages=348–349}}.</ref>
+Expressed more invariantly, the gradient of a vector field {{math|'''f'''}} can be defined by the Levi-Civita connection and metric tensor:
 :<math>\nabla^a f^b = g^{ac} \nabla_c f^b ,</math>
@@ Line 126: / Line 125: @@
 ===Riemannian manifolds===
-For any [[smooth function]] {{mvar|f}} on a Riemannian manifold {{math|(''M'', ''g'')}}, the gradient of {{math|''f''}} is the vector field {{math|∇''f''}} such that for any vector field {{math|''X''}},
+For any smooth function {{mvar|f}} on a Riemannian manifold {{math|(''M'', ''g'')}}, the gradient of {{math|''f''}} is the vector field {{math|∇''f''}} such that for any vector field {{math|''X''}},
 :<math>g(\nabla f, X) = \partial_X f,</math>
 that is,
 :<math>g_x\big((\nabla f)_x, X_x \big) = (\partial_X f) (x),</math>
-where {{math|''g''<sub>''x''</sub>( , )}} denotes the [[inner product]] of tangent vectors at {{math|''x''}} defined by the metric {{math|''g''}} and {{math|∂<sub>''X''</sub>&thinsp;''f''}} is the function that takes any point {{math|''x'' ∈ ''M''}} to the directional derivative of {{math|''f''}} in the direction {{math|''X''}}, evaluated at {{math|''x''}}. In other words, in a [[coordinate chart]] {{math|''φ''}} from an open subset of {{math|''M''}} to an open subset of {{math|'''R'''<sup>''n''</sup>}}, {{math|(∂<sub>''X''</sub>&thinsp;''f''&thinsp;)(''x'')}} is given by:
+where {{math|''g''<sub>''x''</sub>( , )}} denotes the inner product of tangent vectors at {{math|''x''}} defined by the metric {{math|''g''}} and {{math|∂<sub>''X''</sub>&thinsp;''f''}} is the function that takes any point {{math|''x'' ∈ ''M''}} to the directional derivative of {{math|''f''}} in the direction {{math|''X''}}, evaluated at {{math|''x''}}. In other words, in a coordinate chart {{math|''φ''}} from an open subset of {{math|''M''}} to an open subset of {{math|'''R'''<sup>''n''</sup>}}, {{math|(∂<sub>''X''</sub>&thinsp;''f''&thinsp;)(''x'')}} is given by:
 :<math>\sum_{j=1}^n X^{j} \big(\varphi(x)\big) \frac{\partial}{\partial x_{j}}(f \circ \varphi^{-1}) \Bigg|_{\varphi(x)},</math>
 where {{math|''X''{{isup|''j''}}}} denotes the {{math|''j''}}th component of {{math|''X''}} in this coordinate chart.
@@ Line 140: / Line 139: @@
 Generalizing the case {{math|1=''M'' = '''R'''<sup>''n''</sup>}}, the gradient of a function is related to its exterior derivative, since
 :<math>(\partial_X f) (x) = (df)_x(X_x) .</math>
-More precisely, the gradient {{math|∇''f''}} is the vector field associated to the differential 1-form {{math|''df''}} using the [[musical isomorphism]]
+More precisely, the gradient {{math|∇''f''}} is the vector field associated to the differential 1-form {{math|''df''}} using the musical isomorphism
 :<math>\sharp=\sharp^g\colon T^*M\to TM</math>
 (called "sharp") defined by the metric {{math|''g''}}. The relation between the exterior derivative and the gradient of a function on {{math|'''R'''<sup>''n''</sup>}} is a special case of this in which the metric is the flat metric given by the dot product.
 ==Resources==
 * [https://en.wikipedia.org/wiki/Gradient Gradient], Wikipedia

← Older revision		Revision as of 15:51, 19 October 2021
Line 52:		Line 52:

	===Cylindrical and spherical coordinates===		===Cylindrical and spherical coordinates===
−	~~{{main\|Del in cylindrical and spherical coordinates}}~~

	In cylindrical coordinates with a Euclidean metric, the gradient is given by:		In cylindrical coordinates with a Euclidean metric, the gradient is given by:

← Older revision		Revision as of 15:50, 19 October 2021
Line 80:		Line 80:

	The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.		The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.
		+
		+	==Generalizations==
		+
		+	=== Jacobian ===
		+	{{Main\|Jacobian matrix and determinant}}
		+
		+	The [[Jacobian matrix]] is the generalization of the gradient for vector-valued functions of several variables and [[differentiable map]]s between [[Euclidean space]]s or, more generally, [[manifold]]s.<ref>{{harvtxt\|Beauregard\|Fraleigh\|1973\|pp=87,248}}</ref><ref>{{harvtxt\|Kreyszig\|1972\|pp=333,353,496}}</ref> A further generalization for a function between [[Banach space]]s is the [[Fréchet derivative]].
		+
		+	Suppose {{math\|'''f''' : ℝ<sup>''n''</sup> → ℝ<sup>''m''</sup>}} is a function such that each of its first-order partial derivatives exist on {{math\|ℝ<sup>''n''</sup>}}. Then the Jacobian matrix of {{math\|'''f'''}} is defined to be an {{math\|''m''×''n''}} matrix, denoted by <math>\mathbf{J}_\mathbb{f}(\mathbb{x})</math> or simply <math>\mathbf{J}</math>. The {{math\|(''i'',''j'')}}th entry is <math>\mathbf J_{ij} = \frac{\partial f_i}{\partial x_j}</math>. Explicitly
		+
		+	: <math>\mathbf J = \begin{bmatrix}
		+	\dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n} \end{bmatrix}
		+	= \begin{bmatrix}
		+	\nabla^\mathsf{T} f_1 \\
		+	\vdots \\
		+	\nabla^\mathsf{T} f_m
		+	\end{bmatrix}
		+	= \begin{bmatrix}
		+	\dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\
		+	\vdots & \ddots & \vdots\\
		+	\dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix}.</math>
		+
		+	===Gradient of a vector field===
		+	{{see also\|Covariant derivative}}
		+	Since the total derivative of a vector field is a [[linear mapping]] from vectors to vectors, it is a [[tensor]] quantity.
		+
		+	In rectangular coordinates, the gradient of a vector field {{math\|1='''f''' = ( ''f''{{i sup\|1}}, ''f''{{i sup\|2}}, ''f''{{i sup\|3}})}} is defined by:
		+
		+	:<math>\nabla \mathbf{f}=g^{jk}\frac{\partial f^i}{\partial x^j} \mathbf{e}_i \otimes \mathbf{e}_k,</math>
		+
		+	(where the [[Einstein summation notation]] is used and the [[tensor product]] of the vectors {{math\|'''e'''<sub>''i''</sub>}} and {{math\|'''e'''<sub>''k''</sub>}} is a [[dyadic tensor]] of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix:
		+
		+	:<math>\frac{\partial f^i}{\partial x^j} = \frac{\partial (f^1,f^2,f^3)}{\partial (x^1,x^2,x^3)}.</math>
		+
		+	In curvilinear coordinates, or more generally on a curved [[Riemannian manifold\|manifold]], the gradient involves [[Christoffel symbols]]:
		+
		+	:<math>\nabla \mathbf{f}=g^{jk}\left(\frac{\partial f^i}{\partial x^j}+{\Gamma^i}_{jl}f^l\right) \mathbf{e}_i \otimes \mathbf{e}_k,</math>
		+
		+	where {{math\|''g''{{i sup\|''jk''}}}} are the components of the inverse [[metric tensor]] and the {{math\|'''e'''<sub>''i''</sub>}} are the coordinate basis vectors.
		+
		+	Expressed more invariantly, the gradient of a vector field {{math\|'''f'''}} can be defined by the [[Levi-Civita connection]] and metric tensor:<ref>{{harvnb\|Dubrovin\|Fomenko\|Novikov\|1991\|pages=348–349}}.</ref>
		+
		+	:<math>\nabla^a f^b = g^{ac} \nabla_c f^b ,</math>
		+
		+	where {{math\|∇<sub>''c''</sub>}} is the connection.
		+
		+	===Riemannian manifolds===
		+	For any [[smooth function]] {{mvar\|f}} on a Riemannian manifold {{math\|(''M'', ''g'')}}, the gradient of {{math\|''f''}} is the vector field {{math\|∇''f''}} such that for any vector field {{math\|''X''}},
		+	:<math>g(\nabla f, X) = \partial_X f,</math>
		+	that is,
		+	:<math>g_x\big((\nabla f)_x, X_x \big) = (\partial_X f) (x),</math>
		+	where {{math\|''g''<sub>''x''</sub>( , )}} denotes the [[inner product]] of tangent vectors at {{math\|''x''}} defined by the metric {{math\|''g''}} and {{math\|∂<sub>''X''</sub> ''f''}} is the function that takes any point {{math\|''x'' ∈ ''M''}} to the directional derivative of {{math\|''f''}} in the direction {{math\|''X''}}, evaluated at {{math\|''x''}}. In other words, in a [[coordinate chart]] {{math\|''φ''}} from an open subset of {{math\|''M''}} to an open subset of {{math\|'''R'''<sup>''n''</sup>}}, {{math\|(∂<sub>''X''</sub> ''f'' )(''x'')}} is given by:
		+	:<math>\sum_{j=1}^n X^{j} \big(\varphi(x)\big) \frac{\partial}{\partial x_{j}}(f \circ \varphi^{-1}) \Bigg\|_{\varphi(x)},</math>
		+	where {{math\|''X''{{isup\|''j''}}}} denotes the {{math\|''j''}}th component of {{math\|''X''}} in this coordinate chart.
		+
		+	So, the local form of the gradient takes the form:
		+
		+	:<math>\nabla f = g^{ik} \frac{\partial f}{\partial x^k} {\textbf e}_i .</math>
		+
		+	Generalizing the case {{math\|1=''M'' = '''R'''<sup>''n''</sup>}}, the gradient of a function is related to its exterior derivative, since
		+	:<math>(\partial_X f) (x) = (df)_x(X_x) .</math>
		+	More precisely, the gradient {{math\|∇''f''}} is the vector field associated to the differential 1-form {{math\|''df''}} using the [[musical isomorphism]]
		+	:<math>\sharp=\sharp^g\colon T^*M\to TM</math>
		+	(called "sharp") defined by the metric {{math\|''g''}}. The relation between the exterior derivative and the gradient of a function on {{math\|'''R'''<sup>''n''</sup>}} is a special case of this in which the metric is the flat metric given by the dot product.
		+
		+
		+	==Resources==
		+	* [https://en.wikipedia.org/wiki/Gradient Gradient], Wikipedia

@@ Line 24: / Line 24: @@
 * <math>\left. \frac{\partial f}{\partial x}\right|_{x=a}  </math>
 * <math>\left. \frac{\partial f}{\partial \mathbf{x}}\right|_\mathbf{x=a}  </math>
-*<math>\partial_i f</math> and <math>f_{i}</math> : [[Einstein notation]].
+* <math>\partial_i f</math> and <math>f_{i}</math> : Einstein notation.
 ==Definition==