- #1
Telemachus
- 835
- 30
Hi there. I was following a deduction on continuum mechanics for the invariant nature of the first two laws of thermodynamics. The thing is that this deduction works with an identity, and there is something I'm missing to get it.
I have the vector product: ##\vec \omega \times grad \theta##, where θ is a scalar field, ω is a vector field, and ##grad \vec \omega=0##
Now, the book uses this identity: ##\vec \omega \times grad \theta=-curl (\theta \vec \omega)##
The thing is that I've tried to demonstrate the identity, but I couldn't, I get an extra term, and I don't see why it should be zero.
I have that ##\displaystyle curl (\theta \vec \omega)_i=\varepsilon_{ijk}\frac{\partial (\theta \omega_k)}{\partial x_j}=\varepsilon_{ijk} \omega_k \frac{\partial \theta}{\partial x_j}+\varepsilon_{ijk} \theta \frac{\partial \omega_k}{\partial x_j}\rightarrow \displaystyle curl (\theta \vec \omega)=-\vec \omega \times grad \theta+ \theta curl \vec \omega##
So, to get the identity that the book uses I should have that ##\theta curl \vec \omega=0##, and I don't see why that has to be accomplished.
I have the vector product: ##\vec \omega \times grad \theta##, where θ is a scalar field, ω is a vector field, and ##grad \vec \omega=0##
Now, the book uses this identity: ##\vec \omega \times grad \theta=-curl (\theta \vec \omega)##
The thing is that I've tried to demonstrate the identity, but I couldn't, I get an extra term, and I don't see why it should be zero.
I have that ##\displaystyle curl (\theta \vec \omega)_i=\varepsilon_{ijk}\frac{\partial (\theta \omega_k)}{\partial x_j}=\varepsilon_{ijk} \omega_k \frac{\partial \theta}{\partial x_j}+\varepsilon_{ijk} \theta \frac{\partial \omega_k}{\partial x_j}\rightarrow \displaystyle curl (\theta \vec \omega)=-\vec \omega \times grad \theta+ \theta curl \vec \omega##
So, to get the identity that the book uses I should have that ##\theta curl \vec \omega=0##, and I don't see why that has to be accomplished.