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Question regarding unit normal of some function

  1. Dec 8, 2015 #1
    • Moved from a technical forum, so homework template missing.
    Please can you help me with this question.

    A surface h in the three dimensions is given by the equation z=g(x,y)

    Find the unit normal at any point h in two separate ways:

    a)By identifying h with an equipotential for some scalar field φ to be determined and computing ∇φ.

    b)By expressing the points of h in parametric form r(x, y) and computing ∂r/dx × ∂r/dy

    I think that you have to write h(x,y,z)=xi+yj+g(x,y)k
    but can someone explain what a) means



    thanks
     
  2. jcsd
  3. Dec 8, 2015 #2

    SteamKing

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    This article explains the mechanics behind calculating ∇φ and how that can be used to find the unit normal to a surface:

    http://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx

    BTW, ∇φ is called the 'gradient' of the function φ.

    The symbol ∇ is known as an 'operator' and means ∇ = ∂ / ∂x + ∂ / ∂y + ∂ / ∂z

    https://en.wikipedia.org/wiki/Gradient

    Much of this won't make sense if you haven't studied some elementary vector calculus.
     
  4. Dec 9, 2015 #3

    HallsofIvy

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    If z= g(x,y), then [itex]\phi(x,y,z)= z- g(x,y)= 0[/itex], a constant. That is the "equipotential" referred to in (a). Since g is a constant function, it gradient, [itex]\nabla \phi= \frac{\partial \phi}{\partial x}\vec{i}+ \frac{\partial \phi}{\partial y}\vec{j}+ \frac{\partial \phi}{\partial z}\vec{k}[/itex], is perpendicular to surface at every point.
     
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