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Partial Derivative: Finding the vector on a scalar field at point (3,5)

  1. Jan 30, 2012 #1
    1. The problem statement, all variables and given/known data
    A scalar field is given by the function: ∅ = 3x2y + 4y2
    a) Find del ∅ at the point (3,5)
    b) Find the component of del ∅ that makes a -60o angle with the axis at the point (3,5)

    2. Relevant equations
    del ∅ = d∅/dx + d∅/dy

    3. The attempt at a solution
    I completed part a:
    del ∅ = (6xy+4y2)[itex]\hat{i}[/itex] + (3x2+8y)[itex]\hat{j}[/itex] = 120[itex]\hat{i}[/itex] + 67[itex]\hat{j}[/itex] (this answer is correct)

    I am having trouble with part b. My gut feeling is that I need to take a dot product; project the vector from 'part a' onto the vector that makes "a -60o angle with the axis."

    Assuming the equation is |a||b|cosθ,
    • I think a = 120[itex]\hat{i}[/itex] + 67[itex]\hat{j}[/itex]
    • I'm not sure what b is. Maybe a unit vector?
    • I'm thinking θ is 60o + atan(67/120)

    Also, I'm assuming the "axis" that the problem refers to is the x-axis. The answer is -13.02.

    Thank you for any help.

    EDIT: Oops. 120i + 67j is not correct; it's 90i + 67j. |90i + 67j|cos(60+atan(67/90)) = -13.02
    Last edited: Jan 30, 2012
  2. jcsd
  3. Jan 30, 2012 #2


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    Hello luxer5. Welcome to PF !

    [itex]\cos(\theta)\hat{i}+\sin(\theta)\hat{j}[/itex] is a vector that makes an angle of θ with the positive x-axis. It also happens to have a magnitude of 1 .
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