# Taking the second time derivative of a scalar field

Given some scalar field $$\phi(x,y,x,t)$$, taking its first total derivative we get: $$\frac{d\phi}{dt}=\frac{\partial\phi}{\partial t}+\frac{\partial\phi}{\partial x}\frac{dx}{dt}+\frac{\partial\phi}{\partial y}\frac{dy}{dt}+\frac{\partial\phi}{\partial z}\frac{dz}{dt}$$ or: $$\frac{d\phi}{dt}=\frac{\partial\phi}{\partial t}+\vec{v}\cdot \vec{\nabla}\phi$$ But I'm having a hard time simplifying the second derivative: $$\frac{d^2\phi}{dt^2}= \frac{d}{dt}\left(\frac{\partial\phi}{\partial t}\right) + \frac{d}{dt}\left(\vec{v}\cdot \vec{\nabla}\phi\right)$$ $$\frac{d}{dt}\left(\frac{\partial\phi}{\partial t}\right)= \frac{\partial^2\phi}{\partial^2 t}+\vec{v}\cdot \vec{\nabla} \frac{\partial\phi}{\partial t}$$ $$\frac{d}{dt}\left(\vec{v}\cdot \vec{\nabla}\phi\right)=\vec{a}\cdot \vec{\nabla}\phi +\vec{v}\cdot \frac{d}{dt}\vec{\nabla}\phi$$

$$\frac{d^2\phi}{dt^2}= \frac{\partial^2\phi}{\partial^2 t} + \vec{a}\cdot \vec{\nabla}\phi + \vec{v}\cdot\left( \frac{d}{dt}\vec{\nabla}\phi + \vec{\nabla} \frac{\partial\phi}{\partial t}\right)$$ Is there a way to simplify this further? I can't manage to simplify the expression: $$\vec{v}\cdot\left( \frac{d}{dt}\vec{\nabla}\phi + \vec{\nabla} \frac{\partial\phi}{\partial t}\right)$$ Any help would be greatly appreciated!

But I'm having a hard time simplifying the second derivative:...

$$\frac{d^2\phi}{dt^2}= \frac{\partial^2\phi}{\partial^2 t} + \vec{a}\cdot \vec{\nabla}\phi + \vec{v}\cdot\left( \frac{d}{dt}\vec{\nabla}\phi + \vec{\nabla} \frac{\partial\phi}{\partial t}\right)$$

Is there a way to simplify this further?

Not really. You can manipulate it a bit more, but I wouldn't call it simplifying...

I can't manage to simplify the expression: $$\vec{v}\cdot\left( \frac{d}{dt}\vec{\nabla}\phi + \vec{\nabla} \frac{\partial\phi}{\partial t}\right)$$ Any help would be greatly appreciated!

If you want to, you could rewrite: $$\vec{v}\cdot\left( \frac{d}{dt}\vec{\nabla}\phi + \vec{\nabla} \frac{\partial\phi}{\partial t}\right)$$ as $$2\vec{v}\cdot\vec{\nabla} \frac{\partial\phi}{\partial t} +v_i v_j \frac{\partial^2 \phi}{\partial x_i \partial x_j}\;,$$ where the sum over $$i$$ and $$j$$ is implied.

But not sure if you think this is more simplified or less simplified...