Product rule for partial differentiation - Revision history

Vipul: /* Statement for two functions */

2012-04-08T21:38:38Z

Statement for two functions

← Older revision		Revision as of 21:38, 8 April 2012
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			<center>{{#widget:YouTube\|id=lTFCy8V5qDc}}</center>
	===Statement for partial derivatives for functions of multiple variables===		===Statement for partial derivatives for functions of multiple variables===

Vipul: /* Statement for two functions */

2012-04-02T22:52:15Z

Statement for two functions

← Older revision		Revision as of 22:52, 2 April 2012
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	==Statement for two functions==		==Statement for two functions==

	===Statement for partial derivatives===		===Statement for partial derivatives for functions of two variables===

			The derivatives used here are [[partial derivative]]s.

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Version type !! Statement ~~for functions of two variables~~		! Version type !! Statement
	\|-		\|-
	\| specific point, named functions \|\| Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. Suppose <math>(x_0,y_0)</math> is a point in the domain of both <math>f</math> and <math>g</math>. Suppose the partial derivatives <math>f_x(x_0,y_0)</math> and <math>g_x(x_0,y_0)</math> both exist. Let <math>fg</math> denote the [[pointwise product of functions\|product]] of the functions. Then, we have:<br><math>(fg)_x(x_0,y_0) =f_x(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_x(x_0,y_0)</math><br>Suppose the partial derivatives <math>f_y(x_0,y_0)</math> and <math>g_y(x_0,y_0)</math> both exist. Then, we have:<br><math>(fg)_y(x_0,y_0) = f_y(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_y(x_0,y_0)</math>		\| specific point, named functions \|\| Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. Suppose <math>(x_0,y_0)</math> is a point in the domain of both <math>f</math> and <math>g</math>. Suppose the partial derivatives <math>f_x(x_0,y_0)</math> and <math>g_x(x_0,y_0)</math> both exist. Let <math>fg</math> denote the [[pointwise product of functions\|product]] of the functions. Then, we have:<br><math>(fg)_x(x_0,y_0) =f_x(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_x(x_0,y_0)</math><br>Suppose the partial derivatives <math>f_y(x_0,y_0)</math> and <math>g_y(x_0,y_0)</math> both exist. Then, we have:<br><math>(fg)_y(x_0,y_0) = f_y(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_y(x_0,y_0)</math>
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	\|}		\|}

			===Statement for partial derivatives for functions of multiple variables===

			{\| class="sortable" border="1"
			! Version type !! Statement
			\|-
			\| specific point, named functions \|\| Suppose <math>f,g</math> are both functions of variables <math>x_1,x_2,\dots,x_n</math>. Suppose <math>(a_1,a_2,\dots,a_n)</math> is a point in the domain of both <math>f</math> and <math>g</math>. Fix a number <math>i</math> in <math>\{ 1,2,3,\dots,n \}</math>. Suppose the partial derivatives <math>f_{x_i}(a_1,a_2,\dots,a_n)</math> and <math>g_{x_i}(a_1,a_2,\dots,a_n)</math> both exist. Let <math>fg</math> denote the [[pointwise product of functions\|product]] of the functions. Then, we have:<br><math>(fg)_{x_i}(a_1,a_2,\dots,a_n) =f_{x_i}(a_1,a_2,\dots,a_n)g(a_1,a_2,\dots,a_n) + f(a_1,a_2,\dots,a_n)g_{x_i}(a_1,a_2,\dots,a_n)</math>
			\|-
			\| generic point, named functions \|\| Suppose <math>f,g</math> are both functions of variables <math>x_1,_2,\dots,x_n</math>. Then, for any fixed <math>i</math> in <math>\{ 1,2,3,\dots,n \}</math>: <math>(fg)_{x_i}(x_1,x_2,\dots,x_n) =f_{x_i}(x_1,x_2,\dots,x_n)g(x_1,x_2,\dots,x_n) + f(x_1,x_2,\dots,x_n)g_{x_i}(x_1,x_2,\dots,x_n)</math><br>These hold wherever the right side expressions make sense (see [[concept of equality conditional to existence of one side]]).
			\|-
			\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. Then, for any fixed <math>i</math> in <math>\{ 1,2,3,\dots,n \}</math>: <math>(fg)_{x_i} = f_{x_i}g + fg_{x_i}</math><br>These hold wherever the right side expressions make sense (see [[concept of equality conditional to existence of one side]]).
			\|}
	===Statement for directional derivatives===		===Statement for directional derivatives===

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	\|}		\|}

	~~The rule applies at all points where the right side make sense.~~

	===Statement for gradient vectors===		===Statement for gradient vectors===

Vipul: /* Statement for gradient vectors */

2012-04-02T16:02:15Z

Statement for gradient vectors

← Older revision		Revision as of 16:02, 2 April 2012
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	\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0})</math>. Note that the products on the right side are scalar-vector multiplications.		\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0})</math>. Note that the products on the right side are scalar-vector multiplications.
	\|-		\|-
	\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \~~nabla_{~~\overline{u}~~}(fg~~)= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector ''function'' multiplications.		\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla(fg)(\overline{x})= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector ''function'' multiplications.
	\|-		\|-
	\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla(fg) = g\nabla (f) + f\nabla (g)</math>. Note that the products on the right side are scalar-vector ''function'' multiplications.		\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla(fg) = g\nabla (f) + f\nabla (g)</math>. Note that the products on the right side are scalar-vector ''function'' multiplications.

Vipul: /* Statement for gradient vectors */

2012-04-02T04:48:52Z

Statement for gradient vectors

← Older revision		Revision as of 04:48, 2 April 2012
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	\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0})</math>. Note that the products on the right side are scalar-vector multiplications.		\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0})</math>. Note that the products on the right side are scalar-vector multiplications.
	\|-		\|-
	\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector multiplications.		\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector ''function'' multiplications.
	\|-		\|-
	\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla(fg) = g\nabla (f) + f\nabla (g)</math>. Note that the products on the right side are scalar-vector multiplications.		\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla(fg) = g\nabla (f) + f\nabla (g)</math>. Note that the products on the right side are scalar-vector ''function'' multiplications.
	\|}		\|}

Vipul: /* Statement for gradient vectors */

2012-04-02T04:48:22Z

Statement for gradient vectors

← Older revision		Revision as of 04:48, 2 April 2012
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	! Version type !! Statement		! Version type !! Statement
	\|-		\|-
	\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0}) </math>. Note that the products on the right side are scalar-vector multiplications.		\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0})</math>. Note that the products on the right side are scalar-vector multiplications.
	\|-		\|-
	\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector multiplications.		\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector multiplications.

Vipul: /* Statement for directional derivatives */

2012-04-02T04:47:58Z

Statement for directional derivatives

← Older revision		Revision as of 04:47, 2 April 2012
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	! Version type !! Statement		! Version type !! Statement
	\|-		\|-
	\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[directional derivative]]s:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x_0}) = ~~g(\overline{x_0})~~\nabla_{\overline{u}}(f)(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})</math>		\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[directional derivative]]s:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x_0}) = \nabla_{\overline{u}}(f)(\overline{x_0})g(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})</math>
	\|-		\|-
	\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = ~~g(\overline{x})~~\nabla_{\overline{u}}(f)(\overline{x}) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})</math>.		\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = \nabla_{\overline{u}}(f)(\overline{x})g(\overline{x}) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})</math>.
	\|-		\|-
	\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg) = g\nabla_{\overline{u}}(f) + f\nabla_{\overline{u}}(g)</math>.		\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg) = \nabla_{\overline{u}}(f)g + f\nabla_{\overline{u}}(g)</math>.
	\|}		\|}

Vipul: /* Statement for two functions */

2012-04-02T04:46:27Z

Statement for two functions

← Older revision		Revision as of 04:46, 2 April 2012
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	\| specific point, named functions \|\| Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. Suppose <math>(x_0,y_0)</math> is a point in the domain of both <math>f</math> and <math>g</math>. Suppose the partial derivatives <math>f_x(x_0,y_0)</math> and <math>g_x(x_0,y_0)</math> both exist. Let <math>fg</math> denote the [[pointwise product of functions\|product]] of the functions. Then, we have:<br><math>(fg)_x(x_0,y_0) =f_x(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_x(x_0,y_0)</math><br>Suppose the partial derivatives <math>f_y(x_0,y_0)</math> and <math>g_y(x_0,y_0)</math> both exist. Then, we have:<br><math>(fg)_y(x_0,y_0) = f_y(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_y(x_0,y_0)</math>		\| specific point, named functions \|\| Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. Suppose <math>(x_0,y_0)</math> is a point in the domain of both <math>f</math> and <math>g</math>. Suppose the partial derivatives <math>f_x(x_0,y_0)</math> and <math>g_x(x_0,y_0)</math> both exist. Let <math>fg</math> denote the [[pointwise product of functions\|product]] of the functions. Then, we have:<br><math>(fg)_x(x_0,y_0) =f_x(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_x(x_0,y_0)</math><br>Suppose the partial derivatives <math>f_y(x_0,y_0)</math> and <math>g_y(x_0,y_0)</math> both exist. Then, we have:<br><math>(fg)_y(x_0,y_0) = f_y(x_0,y_0)g(x_0,y_0) + f(x_0,y_0)g_y(x_0,y_0)</math>
	\|-		\|-
	\| generic point, named functions \|\| Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. <br><math>(fg)_x(x,y) =f_x(x,y)g(x,y) + f(x,y)g_x(x,y)</math><br><math>(fg)_y(x,y) = f_y(x,y)g(x,y) + f(x,y)g_y(x,y)</math><br>These hold wherever the right side expressions make sense.		\| generic point, named functions \|\| Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. <br><math>(fg)_x(x,y) =f_x(x,y)g(x,y) + f(x,y)g_x(x,y)</math><br><math>(fg)_y(x,y) = f_y(x,y)g(x,y) + f(x,y)g_y(x,y)</math><br>These hold wherever the right side expressions make sense (see [[concept of equality conditional to existence of one side]]).
	\|-		\|-
	\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. <br><math>(f g)_x =f_xg + fg_x</math><br><math>(f g)_y = f_yg + fg_y</math><br>These hold wherever the right side expressions make sense.		\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both functions of variables <math>x,y</math>. <br><math>(f g)_x =f_xg + fg_x</math><br><math>(f g)_y = f_yg + fg_y</math><br>These hold wherever the right side expressions make sense (see [[concept of equality conditional to existence of one side]]).
	\|}		\|}

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	\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[directional derivative]]s:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x_0}) = g(\overline{x_0})\nabla_{\overline{u}}(f)(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})</math>		\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[directional derivative]]s:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x_0}) = g(\overline{x_0})\nabla_{\overline{u}}(f)(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})</math>
	\|-		\|-
	\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = g(\overline{x})\nabla_{\overline{u}}(f)(\overline{x}) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})</math>.		\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = g(\overline{x})\nabla_{\overline{u}}(f)(\overline{x}) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})</math>.
	\|-		\|-
	\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg) = g\nabla_{\overline{u}}(f) + f\nabla_{\overline{u}}(g)</math>.		\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg) = g\nabla_{\overline{u}}(f) + f\nabla_{\overline{u}}(g)</math>.
	\|}		\|}

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	\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0}) </math>. Note that the products on the right side are scalar-vector multiplications.		\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0}) </math>. Note that the products on the right side are scalar-vector multiplications.
	\|-		\|-
	\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg)= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector multiplications.		\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla_{\overline{u}}(fg)= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector multiplications.
	\|-		\|-
	\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense:<br><math>\! \nabla(fg) = g\nabla (f) + f\nabla (g)</math>. Note that the products on the right side are scalar-vector multiplications.		\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):<br><math>\! \nabla(fg) = g\nabla (f) + f\nabla (g)</math>. Note that the products on the right side are scalar-vector multiplications.
	\|}		\|}

Vipul: /* Statement for gradient vectors */

2012-04-02T04:44:18Z

Statement for gradient vectors

← Older revision		Revision as of 04:44, 2 April 2012
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	! Version type !! Statement		! Version type !! Statement
	\|-		\|-
	\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = f(\overline{x_0})\nabla (g)(\overline{x_0}) + g(\overline{x_0}) \nabla (f)(\overline{x_0})</math>. Note that the products on the right side are scalar-vector multiplications.		\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[gradient vector]]s:<br><math>\! \nabla(fg)(\overline{x_0}) = g(\overline{x_0}) \nabla (f)(\overline{x_0}) + f(\overline{x_0})\nabla (g)(\overline{x_0}) </math>. Note that the products on the right side are scalar-vector multiplications.
	\|-		\|-
	\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg)= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector multiplications.		\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Then, we have the following product rule for [[gradient vector]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg)= g(\overline{x}) \nabla (f)(\overline{x}) + f(\overline{x})\nabla (g)(\overline{x})</math>. Note that the products on the right side are scalar-vector multiplications.

Vipul: /* Statement for directional derivatives */

2012-04-02T04:43:39Z

Statement for directional derivatives

← Older revision		Revision as of 04:43, 2 April 2012
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	\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[directional derivative]]s:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x_0}) = g(\overline{x_0})\nabla_{\overline{u}}(f)(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})</math>		\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[directional derivative]]s:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x_0}) = g(\overline{x_0})\nabla_{\overline{u}}(f)(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})</math>
	\|-		\|-
	\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = g(\overline{x})\nabla_{\overline{u}}(f)(\overline{x})) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})</math>.		\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = g(\overline{x})\nabla_{\overline{u}}(f)(\overline{x}) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})</math>.
	\|-		\|-
	\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg) = g\nabla_{\overline{u}}(f) + f\nabla_{\overline{u}}(g)</math>.		\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg) = g\nabla_{\overline{u}}(f) + f\nabla_{\overline{u}}(g)</math>.

Vipul: /* Statement for directional derivatives */

2012-04-02T04:43:06Z

Statement for directional derivatives

← Older revision		Revision as of 04:43, 2 April 2012
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	\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[directional derivative]]s:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x_0}) = g(\overline{x_0})\nabla_{\overline{u}}(f)(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})</math>		\| specific point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Suppose <math>\overline{x_0}</math> is a point in the domain of both functions. Then, we have the following product rule for [[directional derivative]]s:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x_0}) = g(\overline{x_0})\nabla_{\overline{u}}(f)(\overline{x_0}) + f(\overline{x_0})\nabla_{\overline{u}}(g)(\overline{x_0})</math>
	\|-		\|-
	\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = g(\overline{x}\nabla_{\overline{u}}(f)(\overline{x})) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})</math>.		\| generic point, named functions \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg)(\overline{x}) = g(\overline{x})\nabla_{\overline{u}}(f)(\overline{x})) + f(\overline{x})\nabla_{\overline{u}}(g)(\overline{x})</math>.
	\|-		\|-
	\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg) = g\nabla_{\overline{u}}(f) + f\nabla_{\overline{u}}(g)</math>.		\| generic point, named functions, point-free notation \|\| Suppose <math>f,g</math> are both real-valued functions of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{u}</math> is a unit vector. Then, we have the following product rule for [[directional derivative]]s wherever the right side expression makes sense:<br><math>\! \nabla_{\overline{u}}(fg) = g\nabla_{\overline{u}}(f) + f\nabla_{\overline{u}}(g)</math>.