Using logarithms in vector Calculus

[Mzzed]

Homework Statement

My mentor has run me through the derivation of equation (3) bellow. I am unsure how he went from (1) to (3) by incorporating the log term from eq(2). In eq(3) it seems he just canceled the relevant n terms and then identified 1/n as the derivative of L however if this were the case then I am unsure why only one of the two terms on the RHS of eq(3) contains the L term.

How did he get from (1) to (3) using eq(2)?

Homework Equations

1) (∇⋅ωn)/n = (n∇⋅ω + ω⋅∇n)/n
2) L = log(n)
3) ∇⋅ω = ∇⋅ω + ω⋅∇L
where w is a 3 dimensional vector and n is a scalar.

The Attempt at a Solution

I think he may have just identified 1/n = ∇L but then if this were true, where did ∇n go from the second term in eq(1)? Also if this were true the same ∇L term would then be found in the first term wouldn't it?

 

[Orodruin]

Your equation 3 is missing stuff on the left-hand side.

In general for a logarithm you will have $d\log(f) = df/f$.

 

[Mzzed]

I finally realized he used the multivariable chain rule to turn ∇n*1/n into ∇L. Thanks for your help though!