Difference between revisions of "Gradient"

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m (Reverted edits by Mathoreilly (Talk); changed back to last version by Foxtrot)
m (Undo revision 485965 by Karajou (Talk) Okay I think I can find a fix.)
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In [[mathematics]], a '''gradient''' is the rate a [[function]] increases.
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In [[mathematics]], the gradient of a real-valued differentiable function <math>f(x_1,...,x_n)</math> at a point <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum directional derivative of <math>f</math> at <math>p</math>.
  
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If a function has value <math>f(a)</math> at <math>x=a</math> and <math>f(b)</math> at <math>x=b</math> with <math>a<b</math> than the gradient of <math>[a,b]</math> is,
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More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the vector-field:
  
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:<math>\frac{F(b)-F(a)}{b-a}</math>.
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<math>
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\nabla f = (\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n})
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</math>
  
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In schooling, usually early high school, students are taught that the gradient is equal to "the rise over run" or more formally,
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If <math>u</math> is a unit vector in <math>R^n</math>, then, by the chain rule, the directional derivative of <math>f</math> in the direction of <math>u</math> is simply the dot product:
  
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:<math>\mathrm{gradient}=\frac{\mathrm{rise}}{\mathrm{run}}</math>,
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<math>
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\nabla f \cdot u
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</math>
  
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the rise being defined as the difference between the highest point and the lowest point, negative if the highest is on the left and positive if the highest point in on the right. The run is defined as difference of the right <math>x</math>-value and the left <math>x</math>-value (see [[Cartesian coordinates]]).
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Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction <math>u</math> is maximal in the direction of the gradient, and equal to <math>||\nabla f||</math> for <math>u</math> a unit vector in the direction of the gradient.
  
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This concept is usually first introduced with the introduction of [[linear equation]]s. The equation of a straight line in Cartesian coordinates is given by,
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==Properties of the Gradient==
  
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:<math>y=mx+c</math>
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If <math>f</math> is a differentiable function with smooth level sets <math>f^{-1}(c)</math>, then the gradient vector field <math>\nabla f</math> is perpendicular to the level sets of <math>f</math>. For fix a level set <math>S = f^{-1}(c)</math>, and let <math>v</math> be a vector tangent to <math>S</math> at <math>p</math>. Then we can find a curve <math>\gamma(t)</math> on <math>S</math> with <math>\gamma'(0) = v</math>. Now
  
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where <math>m</math> is the gradient of the line and <math>c</math> is the value of the <math>y</math> coordinate when <math>x=0</math>, this is called the [[y-intercept]], e.g, where the line intercepts the [[y-axis]].
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<math>
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f\circ\gamma(t) = c
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</math>
  
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since <math>S</math> is a level set. Taking derivatives of both sides and applying the chain rule, we get that
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<math>
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\nabla f\cdot \gamma'(0) = \nabla f\cdot v = 0
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</math>
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Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.
 
[[Category:mathematics]]
 
[[Category:mathematics]]

Revision as of 23:10, July 2, 2008

In mathematics, the gradient of a real-valued differentiable function at a point is a vector in which points in the direction in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} increases most rapidly at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} . The magnitude of the gradient at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is equal to the maximum directional derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} .

More precisely, we define the gradient, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} to be the vector-field:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f = (\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n}) }

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} is a unit vector in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^n} , then, by the chain rule, the directional derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} in the direction of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} is simply the dot product:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f \cdot u }

Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} is maximal in the direction of the gradient, and equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ||\nabla f||} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} a unit vector in the direction of the gradient.

Properties of the Gradient

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a differentiable function with smooth level sets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(c)} , then the gradient vector field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f} is perpendicular to the level sets of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} . For fix a level set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = f^{-1}(c)} , and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} be a vector tangent to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} . Then we can find a curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(t)} on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma'(0) = v} . Now

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\circ\gamma(t) = c }

since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a level set. Taking derivatives of both sides and applying the chain rule, we get that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f\cdot \gamma'(0) = \nabla f\cdot v = 0 }

Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla f} is perpendicular to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , i.e., the gradient of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is perpendicular to the level sets of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} .

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