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Gradient and finding the direction of maximum rate of change

  1. Mar 28, 2017 #1
    1. The problem statement, all variables and given/known data
    Hi guys, it a very simple question, but it causing me a great deal of confusion. The questions are as follows:

    upload_2017-3-28_10-58-20.png
    So I worked out the ans for one which I have displayed below. But what I don't understand is what they want from the second question. Because the way I see it, the direction of maximum rate of change is the gradient itself. Am I missing something here because I am really lost.

    2. Relevant equations

    3. The attempt at a solution
    1: ##\nabla f=\frac{-xi}{(x^2+y^2+z^2)^3} -\frac{yj}{(x^2+y^2+z^2)^3}-\frac{zk}{(x^2+y^2+z^2)^3}##

    ##\nabla f = le^{lx+my+nz}i+me^{lx+my+nz}j+ne^{lx+my+nz}k##

    2: I make the direction as follows: ##(-xi-yj-zk)## & ##li+mj+nk##

    Is this correct or have I miss understood the question?
     
  2. jcsd
  3. Mar 28, 2017 #2

    BvU

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    Check the exponent in the denominator in the first f in part 1).
    For part 2: I agree with you : ##-\bf\hat r## and ##(l,m,n)##.

    [edit] perhaps for 'direction' the exercise composer wants to see ##\bf \hat r## ?
     
  4. Mar 28, 2017 #3

    PeroK

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    The direction of maximum rate of change is the direction of the gradient. That generally means a unit vector,
     
  5. Mar 28, 2017 #4
    Thank you I did not put the 3/2 in thanks.
     
  6. Mar 28, 2017 #5
    when u say unit vector you mean ##\frac{\nabla f}{|\nabla f|}##?
     
  7. Mar 28, 2017 #6

    PeroK

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    In this case, yes. If you look at the difference between the gradient and the direction if the gradient for the second function - the exponential - you'll see the point.
     
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