gradient facts
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A brief set of Phil's notes (signed PhL, dated 1.31.03) listing lemmas, theorems and short proofs for functions of the radius r. They cover the gradient of f(r), the curl of f(r) r being zero, the divergence of f(r) r as 3f + r f', a related derivative identity, and a curl-of-curl style result, each with applications such as f = 1/r. Some symbols are lost in the text extraction.
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Some Gradient Facts PhL 1.31.03
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Lemma: (r) = Proof: (r/xj ) = 1/2 * 1/r * 2 xj = xj/r = j
Theorem: ( f(r) ) = f '(r) Proof: f/xj = ( f/r) (r/xj ) = f '(r) j
Applications:
f=f ( f(r) ) = f '(r)
f=r (r) =
f=1/r (1/r) = - /r2
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Lemma: x r = 0 Proof: ijkjrk = ijkjk = ijj = 0.
Theorem: x [ f(r) r ] = 0
Proof: x [ f(r) r ] = ( f(r)) x r + f(r) x r = f '(r) x r + f(r) x r = 0 + 0 = 0
Applications: x = 0
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Lemma: r = 3 jrj = 1+1+1 = 3
Theorem: [ f(r) r ] = 3 f(r) + r f '(r)
Proof: ( f(r) r) = r ( f(r) ) + f r = r f '(r) + 3 f = 3 f + r f '(r)
Applications: f = f [ f r ] = 3 f + r f '
f = 1 r = 3
f = 1/r = 2/r
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Theorem: (k) [ f(r) r ] = f(r) k + r f '(r) (k)
Proof: (k) [ f(r) r ] = kii [ f(r) rj ] = ki [ f(r) ij + rj i f(r)] = f(r) k + [k(f(r)) ] r
= f(r) k + [k f '(r) ] r
Applications: f = f (k) [ f r ] = f k + r f ' (k)
f = 1: (k) r = k
f = 1/r: (k) = (1/r) [ k - (k ) ]
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Theorem: x [f(r) r ] x k = -2 k f(r) +r f '(r) { -k + (k) }
Proof: x [f(r) r ] x k = - k ([f(r) r]) + (k) [f(r) r]
= -k [ 3 f(r) + r f '(r) ] + [ f(r) k + (k) [r f '(r)] ] = -2 k f(r) +r f '(r) { -k + (k) }
Applications: f = f x [f r ] x k = -2 k f +r f ' { -k + (k) }
f = 1 x r x k = -2 k
f = 1/r x x k = - (1/r) [ k + (k) ]
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