Yahoo Answers: Answers and Comments for Divergence of unit vector r(hat)? [Physics]
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From chemistry student
enUS
Wed, 24 Oct 2012 13:13:25 +0000
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Yahoo Answers: Answers and Comments for Divergence of unit vector r(hat)? [Physics]
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From Nick: r(hat) = r/r = xi + yj + zk/r
r = √(x...
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Fri, 26 Oct 2012 06:16:01 +0000
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.

From obelix: r(hat) is NOT i+j+k
r(hat) = x/sqrt(x^2 + y^2 ...
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Wed, 24 Oct 2012 15:16:29 +0000
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_coordinates), 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

From mcglamery: Vectors are a geometrical way of representing ...
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Wed, 28 Dec 2016 10:18:13 +0000
Vectors are a geometrical way of representing portions that have direction besides as cost. An occasion of a vector is tension. If we are to totally describe a tension on an merchandise we could desire to specify no longer in basic terms how lots tension is utilized, yet additionally wherein direction. yet another occasion of a vector quantity is speed  an merchandise this is travelling at ten meters according to 2d to the east has a various speed than an merchandise this is travelling ten meters according to 2d to the west. This vector is a particular case, besides the incontrovertible fact that,each so often each physique is drawn to easily the cost of the cost of an merchandise. This quantity, a scalar, is noted as speed which has cost yet no given direction. while vectors are written, they're represented by using a single letter in ambitious form or with an arrow above the letter, jointly with or . some examples of vectors are displacement (e.g. one hundred twenty cm at 30°) and speed (e.g. 12 meters according to 2d north). the only userfriendly SI unit this is a vector is the meter. All others are scalars. Derived portions could nicely be vector or scalar, yet each vector quantity could desire to contain meters in its definition and unit.

From Ben: I dont know how you got 2/r because divergence...
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Wed, 24 Oct 2012 13:19:56 +0000
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.