Find grad *((bbr)/(r^2)) where bbr(x,y,z)=x bbi+y bbj+z bbk, r=sqrt(x^2+y^2+z^2) In this case, why can we not bring the scalar function r^(-2) outside the dot product?

Ayanna Jarvis 2022-10-23 Answered
Find ( r r 2 ) where r ( x , y , z ) = x i + y j + z k, r = x 2 + y 2 + z 2
In this case, why can we not bring the scalar function r 2 outside the dot product?
When I leave the function in place, I calculate a result of 1 r 2 . When I pull it out, I find 3 r 2
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Answers (2)

cokeman206
Answered 2022-10-24 Author has 18 answers
It's not just a dot product, it's a divergence operator, and the variables x,y,z appear in r 2 , it's not a constant.
However, you may use the product rule:
( ψ F ) = ψ F + ψ F
And with ψ = r 2 , it's not difficult to prove that
ψ = 2 r 4 r
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faois3nh
Answered 2022-10-25 Author has 4 answers
( r r 2 ) = ( x x 2 + y 2 + z 2 i + y x 2 + y 2 + z 2 j + z x 2 + y 2 + z 2 k ) = y 2 + z 2 x 2 ( x 2 + y 2 + z 2 ) 2 + x 2 + z 2 y 2 ( x 2 + y 2 + z 2 ) 2 + x 2 + y 2 z 2 ( x 2 + y 2 + z 2 ) 2 = x 2 + y 2 + z 2 ( x 2 + y 2 + z 2 ) 2 = 1 x 2 + y 2 + z 2 = 1 r 2 .
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