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Vector calculus - How to use the gradient?

  1. May 26, 2015 #1
    • Member warned about posting without the template
    I have done part A (i think) really not sure where to begin with the rest of the parts, would appreciate a tip in the right direction, its revision for my first year physics exams in a few weeks.

    Consider the funtion T in the plane (x,y), given by T=ln(x^2 + y^2)

    at point 1,2

    a) in which direction is most rapid increase in T

    I did Grad(T) to get a vector which i think is in the direction of most rapid increase (2/5,4/5)

    b) what distance in this direction gives an inrease of .2 in T

    c) what distance in direction i + j gives and increase of .12 in T

    d) in what directions will T be stationary.

    I dont want solutions, just how to go about solving the problems
     
  2. jcsd
  3. May 26, 2015 #2
    I have done part A (i think) really not sure where to begin with the rest of the parts, would appreciate a tip in the right direction, its revision for my first year physics exams in a few weeks.

    Consider the funtion T in the plane (x,y), given by T=ln(x^2 + y^2)

    at point 1,2

    a) in which direction is most rapid increase in T

    I did Grad(T) to get a vector which i think is in the direction of most rapid increase (2/5,4/5)

    b) what distance in this direction gives an inrease of .2 in T

    c) what distance in direction i + j gives and increase of .12 in T

    d) in what directions will T be stationary.

    I dont want solutions, just how to go about solving the problems
     
  4. May 26, 2015 #3

    Ray Vickson

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    For (b): if you go along direction (2/5,4/5) from the point (1,2) you are looking at points of the form ##x = x(t) = 1 + (2/5)t, \:y = y(t) = 2 + (4/5)t##, where ##t > 0## is a scalar.
     
  5. May 29, 2015 #4
    (DelT = delta T (change in T))

    Turns out the best way to do it for those who are interested is you use DelT = GradT . r, to get the largest change in t (highest delT) r and GradT must be in the same direction. To work out how far in a certain direcction it changes by a certain amount, set delT to whatever you want the change to be (.2 for b) then set r to be a vecctor with magnitude a and direction the same as the direction it was in a, then simply solve for a. Do the same for part c, and finally for part d, set delT to 0 so GradT must be perpendicular to r, which is pretty easy to do by inspection.

    Thanks for the reply.
     
  6. May 29, 2015 #5
    For simplicity, direction of (2/5,4/5) is the same of (1,2).
     
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