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Math Help - div F and curl F

  1. #1
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    div F and curl F

    Quick question:

    A quest states:

    In exercises 1-11, calculate div F and curl F for the given vector fields.

    11. F = θ̂ (with a "hat" on top) = -sinθi + cosθj.

    If I rewrite F in terms of cartesian coordinates I get:

    -(y/(
    √(x2 + y2)) + (x/√(x2 + y2))

    Then by differentiation followed up by addition as the devergence theorem says I get anything but 0, which is the correct result for this task. How come? Have I not converted the polar coordinates into cartesian coordinates correctly? If I can solve this problem, finding the curl of F shouldn't be that hard.
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  2. #2
    MHF Contributor
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    Re: div F and curl F

    Quote Originally Posted by kaemper View Post
    Quick question:

    A quest states:

    In exercises 1-11, calculate div F and curl F for the given vector fields.

    11. F = θ̂ (with a "hat" on top) = -sinθi + cosθj.

    If I rewrite F in terms of cartesian coordinates I get:

    -(y/(
    √(x2 + y2)) + (x/√(x2 + y2))

    Then by differentiation followed up by addition as the devergence theorem says I get anything but 0, which is the correct result for this task. How come? Have I not converted the polar coordinates into cartesian coordinates correctly? If I can solve this problem, finding the curl of F shouldn't be that hard.
    $F=\{ \dfrac{-y}{\sqrt{x^2+y^2}}, \dfrac{x}{\sqrt{x^2+y^2}},0 \}$

    $\dfrac{\partial}{\partial x}F_x=\dfrac{x y}{\left(x^2+y^2\right)^{3/2}}$

    $\dfrac{\partial}{\partial y}F_y=\dfrac{-x y}{\left(x^2+y^2\right)^{3/2}}$

    $\dfrac{\partial}{\partial z}F_z=0$

    $\nabla\cdot F=\dfrac{x y}{\left(x^2+y^2\right)^{3/2}}+\dfrac{-x y}{\left(x^2+y^2\right)^{3/2}}+0=0$
    Thanks from kaemper
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