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How to find parametric equations for three level curves?

  1. Oct 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Find parametric equations for the three level curves of the function
    W(x,y) = sin(x) e^y
    which pass through the points P = (0,1), Q = (pi/2, 0) and R = (pi/6, 3)
    Also compute the vectors of the gradient vector field (gradient of W) at the points P, Q an R

    2. Relevant equations
    Gradient W = (∂W/dx) i + (∂W/dy) j


    3. The attempt at a solution
    Finding level curves:
    At P: W(0,1) = sin(0) e^1 = 0
    sin(x) e^y = 0
    => sin(x) = 0 => x = nπ, where nεZ
    and y ε R
    Level curves are
    x = nπ (where n ε Z)
    y = t (where t ε R)

    At Q: W(∏/2,0) = sin(∏/2) e^0 = 1
    sin(x) e^y = 1
    sin(x) ≠ 0 => x ≠ n∏(where n ε Z)
    Also, e^y = csc(x)
    Level curves are
    x = t (where t ε R, t ≠ n∏, n ε Z)
    y = csc(t)

    At R: W(∏/6, 3) = sin(∏/6) e^3 = (1/2) e^3
    sin(x) e^y = (1/2) e^3
    sin(x) ≠ 0 => x ≠ n∏(where n ε Z)
    Also, e^y = (1/2) e^3 csc(x)
    Level curves are
    x = t (where t ε R, t ≠ n∏, n ε Z)
    y = (1/2) e^3 csc(t)

    Finding gradient:
    ∂W/∂x = cos(x) e^y
    ∂W/dy = sin(x) e^y
    Grad W= cos(x) e^y i + sin(x) e^y j
    At P: Grad W = e i

    At Q: Grad W = j

    At R: Grad W = (√3 /2) e^3 i + (1/2) e^3 j

    Is the above correct?
     
  2. jcsd
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