Divergence of unit vector r(hat)?

r=xi+yj+zk

(r, i, j, and k are vectors)

Evaluate Del ( r/ |r| )

The answer should be 2/r but I cannot get it.

Any help is appreciated!

Update:

2/r is the answer in the back of the book.

Since ( r / |r| ) is the unit vector r(hat), x, y, and j would each equal 1.

So r(hat) = i+j+k

When I take partial derivatives with respect to x, y, and z respectively, I should get 0 for everything. But the answer is 2 / r.

4 Answers

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  • Nick
    Lv 6
    7 years ago
    Best answer

    r(hat) = r/|r| = xi + yj + zk/|r|

    |r| = √(x^2 + y^2 + z^2)

    => r(hat) = (x/√(x^2 + y^2 + z^2))i + (y/√(x^2 + y^2 + z^2))j + (z/√(x^2 + y^2 + z^2))k

    Not i + j + k as you have assumed. Each component of r(hat) has x,y,z dependence.

    =>∇.r(hat) = ∂(x/√(x^2 + y^2 + z^2))/∂x + ∂(y/√(x^2 + y^2 + z^2))/∂y + ∂(z/√(x^2 + y^2 + z^2))/∂z

    Now:

    ∂(x/√(x^2 + y^2 + z^2))/∂x = (√(x^2 + y^2 + z^2) - x^2/√(x^2 + y^2 + z^2)) / (x^2 + y^2 + z^2)

    ∂(y/√(x^2 + y^2 + z^2))/∂x = (√(x^2 + y^2 + z^2) - y^2/√(x^2 + y^2 + z^2)) / (x^2 + y^2 + z^2)

    ∂(z/√(x^2 + y^2 + z^2))/∂x = (√(x^2 + y^2 + z^2) - z^2/√(x^2 + y^2 + z^2)) / (x^2 + y^2 + z^2)

    by the quotient rule of differentiation. Add these together to give:

    ∇.r(hat) = 3/√(x^2 + y^2 + z^2) - (x^2 + y^2 + z^2)/(x^2 + y^2 + z^2)^(3/2)

    => ∇.r(hat) = 3/√(x^2 + y^2 + z^2) - 1/√(x^2 + y^2 + z^2)

    => ∇.r(hat) = 2/√(x^2 + y^2 + z^2)

    => ∇.r(hat) = 2/|r|

    remember to distinguish between r (the vector) and |r| = √(x^2 + y^2 + z^2) (the magnitude of the vector).

    If you like you can copy the array of different mathematical symbols I have on my profile to yours, it makes things easier when asking a mathematics based question.

    Hope this was of some use.

  • 3 years ago

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  • Ben
    Lv 4
    7 years ago

    I dont know how you got 2/r because divergence is calculated by taking the partial with respect to x of the i components, partial with respect to y for the j components and the partial with respect to z under the k components, your answer will be a scalar value which will tell you how the flow is passing a specific point in your field, ie, is it getting denser or is the fluid leaving faster than it is entering.

  • obelix
    Lv 5
    7 years ago

    r(hat) is NOT i+j+k

    r(hat) = x/sqrt(x^2 + y^2 + z^2)i + y/|r| j + z/|r| k

    |r| being, of course, sqrt(x^2 + y^2 + z^2)

    nor you try the problem

    Div(r/|r|) = d/dx (x/sqrt(x^2 + y^2 + z^2)) + d/dy (y/|r|) + d/dz (z/|r|), and remember d/dx is really del/delx meaning they are partial differentials ... now go ahead with the differentiation and probably you'll get the answer

    better yet, use the formula for divergence in spherical coordinates (http://en.wikipedia.org/wiki/Divergence#Spherical_... then you get the answer in one line :

    essentially it will boil down to : Div(r/|r|) = 1/r^2 . d/dr (r^2) = 1/r^2 . 2r = 2/r

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