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Direction of the gradient

  1. Jan 1, 2008 #1
    Hi. The book I'm reading says "We define the vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar as the gradient of that scalar". But how does one know in which direction the maximum increase is?
     
  2. jcsd
  3. Jan 1, 2008 #2

    mda

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    I suggest you consult standard sources to see how the gradient is calculated. Wikipedia is usually a good start.
     
  4. Jan 1, 2008 #3
    I know how to calculate it. Then I guess my qustion is why the gradient is pointing in the direction of maximum increase.
     
  5. Jan 1, 2008 #4
    gradient involves dot product, and cos is maximized when the angle is 0. That gives the direction for maximum rate.
     
  6. Jan 1, 2008 #5

    Vid

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    Start from the directional derivative. grad(f)*u. This is equal to |grad(f)||u|cos(x), where x is the angle between them. This is maximum when cos(x) = 1 which occurs when x = 0. Implying that the maximum rate of change is in the direction of the gradient itself.
     
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