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Math Help - Vector Derivatives

  1. #1
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    Vector Derivatives

    Given a vector field V(x,y,z) = (xi + yj + zk)/(r^3) where r = sqrt(x^2 + y^2 + z^2)
    a) what is the x,y,z components of V(x,y,z)
    b) partial derivatives of each component

    do i substitute the r^3 with the sqrt thing or i just leave it as r^3 when i find the x,y,z components and derivatives?
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  2. #2
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    Quote Originally Posted by AesTheBroken View Post
    Given a vector field V(x,y,z) = (xi + yj + zk)/(r^3) where r = sqrt(x^2 + y^2 + z^2)
    a) what is the x,y,z components of V(x,y,z)
    b) partial derivatives of each component

    do i substitute the r^3 with the sqrt thing or i just leave it as r^3 when i find the x,y,z components and derivatives?

    I think you can safely leave as r^3: as long as it is clear what it is who cares?

    Now, r^3=\left(x^2+y^2+z^2\right)^{3\slash 2}\Longrightarrow \frac{d(r^3)}{dx}=3x\sqrt{x^2+y^2+z^2} , and the same with the other two variables but instead 3x we have 3y,\,3z , resp. , so:

    V(x,y,z)=\left(\frac{x}{r^3},\frac{y}{r^3},\frac{z  }{r^3}\right) \Longrightarrow \frac{dV}{dx}=\left(\frac{r^3-x\frac{d(r^3)}{dx}}{r^6},\frac{-y\frac{d(r^3)}{dy}}{r^6},\frac{-z\frac{d(r^3)}{dz}}{r^6}\right) .

    Tonio
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  3. #3
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    if i wanna find the curl and divergence of v(x y z)? leave it as r too?
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