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Directional derivative with angle

  1. Feb 18, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the directional derivative of the function given below as you arrive at (0,1,3) from the direction of (1,1,0). Give an exact answer.
    f(x,y,z) = sin(5 x) + ln(y^2+1) + z^3

    2. Relevant equations



    3. The attempt at a solution
    I know how to solve this if it were a 2 dimension, I am just confused on what should I multiply the k direction with? The i is multiplied with cos, the j is with sin and what about the k?
     
  2. jcsd
  3. Feb 18, 2009 #2

    Dick

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    Use grad(f) again. The directional derivative of f in the direction n is grad(f).n (dot product, as usual).
     
  4. Feb 18, 2009 #3
    dot product with the vector? how do you get the vector here?
     
  5. Feb 18, 2009 #4

    Dick

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    You are GIVEN the direction vector. It's (1,1,0). Seems hard to read the problem any other way.
     
  6. Feb 18, 2009 #5
    and so what do I use the (0,1,3) for?
     
  7. Feb 18, 2009 #6

    Dick

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    Don't you need a point to evaluate the gradient at?
     
  8. Feb 18, 2009 #7
    ok so the derivative I found is 5*cos(5x)i + 2y/(y^2+1)j + (3z^2)k and after plugging in the points I have j + 27k, so I need to take the dot product of this with i + j ?
     
  9. Feb 18, 2009 #8
    Not to confuse anyone but maybe i'm confusing my self.. just to check

    "Find the directional derivative of the function given below as you arrive at (0,1,3) from the direction of (1,1,0). Give an exact answer.
    f(x,y,z) = sin(5 x) + ln(y^2+1) + z^3"


    it says FROM the direction of (1,1,0)

    wouldn't that mean calculate the gradient at the point (1,1,0).

    and the direction vector would be the in the direction so you would ARRIVE at (0,1,3).

    in other words

    V = (0 - 1, 1 - 1, 3 - 0) = (-1,0,3)
    unit vector = [(-1,0,3)] /root(10)



    so calculate

    gradient at (1,1,0) dot [(-1,0,3)]/root(10)
     
  10. Feb 18, 2009 #9

    Dick

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    5*cos(5*0) isn't 0. You have a habit of asking questions that have already been answered. In post 2, I said find grad(f).n, and now you are asking if you should take the dot product again? Why?
     
  11. Feb 18, 2009 #10
    Therefore it is, 5i + j + 27k and I need to take a dot product of this with i + j and resulting in 6
     
  12. Feb 18, 2009 #11

    Dick

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    That's what I think. But I'm still scratching my head over salman213's suggestion. What do you think?
     
  13. Feb 18, 2009 #12
    And that's the reason why I am asking again and again, is to make sure that it is right.. as it seems like the other way around as it says from the direction
     
  14. Feb 18, 2009 #13
    looking at salman213 suggestion i get (-5cos(5))/sqrt(10), and I tried that but the result is wrong.... I don't know if I miscalculate anything
     
  15. Feb 18, 2009 #14

    djeitnstine

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    The question seems ambiguous but I do agree with Dick on this. it says from the direction of 1,1,0 not from the POINT 1,1,0. So when one is arriving at some place usually you arrive at a point. In this case you are arriving from a particular direction -as the statement says- and not another point.
     
  16. Feb 18, 2009 #15
    I tried 6 and it gives me a wrong result as well.... weird
     
  17. Feb 18, 2009 #16

    Dick

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    Well, you are confusing people apparently. How can you 'arrive' at (0,1,3) after 'starting' at (1,1,0). That might make a little sense if that were on the same level curve, in which case the answer would be 0. But they aren't. How else do you 'get from' (1,1,0) to (0,1,3)? You might interpret it as 'find the directional derivative of f along the line from (1,1,0) to (0,1,3)', but I don't think so. What's the magnitude of the vector to find the directional derivative along, unit vector? I think you are throwing us into muddy water. I'd stick with the simple interpretation.
     
  18. Feb 18, 2009 #17
    The question it self is confusing and that's why I asked it here....
     
  19. Feb 18, 2009 #18

    Dick

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    Oh, just great. Do they mean a unit vector along the along the line connecting (1,1,0) to (0,1,3)?
     
  20. Feb 18, 2009 #19
    I guess it is, as 6 doesn't work...
     
  21. Feb 18, 2009 #20

    Dick

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    I'll grant you that in retrospect.
     
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