- #1
fluidistic
Gold Member
- 3,924
- 261
At some point, in Physics (more precisely in thermodynamics), I must take the divergence of a quantity like ##\mu \vec F##. Where ##\mu## is a scalar function of possibly many different variables such as temperature (which is also a scalar), position, and even magnetic field (a vector field).
My question is, how to evaluate that divergence? I am tempted to set it equal to ##\nabla \cdot (\mu \vec F)=\nabla \mu \cdot \vec F + \mu \nabla \cdot \vec F##. I know it doesn't matter here, but it turns out that thanks to some physical fact, the divergence of ##\vec F## vanishes, so we can focus solely on the first term if we want.
And that is where my doubt lies. Precisely, the gradient of ##\mu##. Is it like a total derivative? So that if ##\mu## depends on temperature, magnetic field and position, then I should evaluate ##\nabla \mu## as ##\left ( \frac{\partial \mu}{\partial T}\right)_{\vec B,x}\frac{\partial T}{\partial x} + \left ( \frac{\partial \mu}{\partial x}\right)_{\vec B,T}\frac{\partial x}{\partial x} + \left ( \frac{\partial \mu}{\partial B_x}\right)_{B_y, B_z,x,T}\frac{\partial B_x}{\partial x}+... ##? I am a bit confused on the number of terms and whether what I wrote is correct.
My question is, how to evaluate that divergence? I am tempted to set it equal to ##\nabla \cdot (\mu \vec F)=\nabla \mu \cdot \vec F + \mu \nabla \cdot \vec F##. I know it doesn't matter here, but it turns out that thanks to some physical fact, the divergence of ##\vec F## vanishes, so we can focus solely on the first term if we want.
And that is where my doubt lies. Precisely, the gradient of ##\mu##. Is it like a total derivative? So that if ##\mu## depends on temperature, magnetic field and position, then I should evaluate ##\nabla \mu## as ##\left ( \frac{\partial \mu}{\partial T}\right)_{\vec B,x}\frac{\partial T}{\partial x} + \left ( \frac{\partial \mu}{\partial x}\right)_{\vec B,T}\frac{\partial x}{\partial x} + \left ( \frac{\partial \mu}{\partial B_x}\right)_{B_y, B_z,x,T}\frac{\partial B_x}{\partial x}+... ##? I am a bit confused on the number of terms and whether what I wrote is correct.