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Math Help - Divergence Theorem, vector fields ..........

  1. #1
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    Divergence Theorem, vector fields ..........

    Hello,
    plz try to do these questions.
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  2. #2
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    Let me try...

    First we need to find,
    \mbox{div} \b{F}=2x-1
    And S is a box.
    Thus,
    \int_0^3 \int_0^1 \int_0^2 2x-1 dz\, dy\, dx
    Thus,
    \int_0^3 \int_0^1 4x-2 dy\, dx
    Thus,
    \int_0^3 4x-2 dx
    Thus,
    2(3)^2=18
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  3. #3
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    Question 2]

    You are given,
    \b{V}=<e^x\cos y,-e^x\sin y,0>
    To determine if it is conservative for well-behaved vector fields the necessary and sufficient conditions is to check if \mbox{curl}\b{F}=\b{0}
    Thus, the we have,
    \left| \begin{array}{ccc}\bold{i}&\bold{j}&\bold{k}\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ e^x\cos y&-e^x\sin y&0  \end{array} \right|=<0,0,-e^x\sin y+e^x\sin y>=\b{0}
    Thus if is.

    It turns out that there is an easier way in the special case for two dimension vector fields.
    \b{F}(x,y)=u(x,y)\b{i}+v(x,y)\b{j}
    Necessary and suffient for well-behaves fields,
    u_y(x,y)=v_x(x,y)
    Called the "cross-partials test"
    (If you taken Differencial Equations you will note it is the same test to determine if a differencial equation is exact or not).
    In this case,
    u_y=-e^x\sin y
    v_x=-e^x\sin y
    Thus, it is conservative.
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