Vipul: Created page with "==Statement of relation== {| class="sortable" border="1" ! Version type !! Statement |- | at a point, in vector notation (multiple variables) || Suppose $f$ is a f..."

2012-05-04T15:20:56Z

Created page with "==Statement of relation== {| class="sortable" border="1" ! Version type !! Statement |- | at a point, in vector notation (multiple variables) || Suppose <math>f</math> is a f..."

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==Statement of relation==

{| class="sortable" border="1"
! Version type !! Statement
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| at a point, in vector notation (multiple variables) || Suppose <math>f</math> is a function of a vector variable <math>\overline{x} = \langle x_1,x_2,\dots,x_n \rangle</math>. Suppose <math>\overline{u}</math> is a unit vector and <math>\overline{a}</math> is a point in the domain of <math>f</math>. Suppose that the [[gradient vector]] of <math>f</math> at <math>\overline{a}</math> exists. We denote this gradient vector by <math>\nabla f(\overline{a})</math>. Then, we have the following relationship: <math>D_{\overline{u}}(f)(\overline{a}) = \overline{u} \cdot (\nabla f(\overline{a}))</math> The right side here is the [[dot product of vectors]].
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| generic point, in vector notation (multiple variables) || Suppose <math>f</math> is a function of a vector variable <math>\overline{x} = \langle x_1,x_2,\dots,x_n \rangle</math>. Suppose <math>\overline{u}</math> is a unit vector. We then have: <math>D_{\overline{u}}(f)(\overline{x}) = \overline{u} \cdot (\nabla f(\overline{x}))</math> The right side here is a [[dot product of vectors]]. The equality holds [[concept of equality conditional to existence of one side|whenever the right side makes sense]].
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| generic point, point-free notation (multiple variables) || Suppose <math>f</math> is a function of a vector variable <math>\overline{x} = \langle x_1,x_2,\dots,x_n \rangle</math>. Suppose <math>\overline{u}</math> is a unit vector. We then have: <math>D_{\overline{u}}(f) = \overline{u} \cdot (\nabla f)</math> The right side here is a dot product of vector-valued functions (the constant function <math>\overline{u}</math> and the gradient vector of <math>f</math>). The equality holds [[concept of equality conditional to existence of one side|whenever the right side makes sense]].
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Relation between gradient vector and directional derivatives - Revision history

Vipul: Created page with "==Statement of relation== {| class="sortable" border="1" ! Version type !! Statement |- | at a point, in vector notation (multiple variables) || Suppose f is a f..."

Vipul: Created page with "==Statement of relation== {| class="sortable" border="1" ! Version type !! Statement |- | at a point, in vector notation (multiple variables) || Suppose $f$ is a f..."