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Line integrals, gradient fields

  1. Aug 1, 2014 #1
    1. The problem statement, all variables and given/known data

    ##\nabla{F} = <2xyze^{x^2},ze^{x^2},ye^{x^2}##
    if f(0,0,0) = 5 find f(1,1,2)


    2. Relevant equations



    3. The attempt at a solution

    my book doesn't have a good example of a problem like this, am I looking for a potential?

    ##<\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z} >= <2xyze^{x^2},ze^{x^2},ye^{x^2}>##
     
    Last edited: Aug 1, 2014
  2. jcsd
  3. Aug 1, 2014 #2

    CAF123

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    Gold Member

    Yes, solve the three differential equations to find F(x,y,z) up to a constant. The given condition sets this constant, from which you can find F(x,y,z) for all x,y,z.
     
  4. Aug 1, 2014 #3


    ##\frac{\partial}{\partial x} = 2xyze^{x^2} \Rightarrow \int 2xyze^{x^2}dx = yze^{x^2} + h(y,z)##

    ##\frac{\partial}{\partial y} = ze^{x^2} = ze^{x^2} + h(y,z) \Rightarrow h_{y}(y,z) = 0##

    ##\frac{\partial}{\partial z} = ye^{x^2} = ye^{x^2}+h(z) \Rightarrow h_{z}(z) = h'(z)##

    ##h'(z) = 0 \Rightarrow h(z) = k## for a constant k, and because f(0,0,0) = 5 then

    ##f= yze^{x^2} + 5##

    so ##f(1,1,2) = 1(2)e^1+5 = 2e+5##
     
    Last edited: Aug 1, 2014
  5. Aug 1, 2014 #4

    ehild

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    Homework Helper
    Gold Member

    Correct!

    ehild
     
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