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user7530
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# Material Derivative of the Gradient of a Scalar Field

Let $$f$$ be a scalar field that is continuous and does not vary along the flow, that is $$D_t(f)=0$$ where $$D_t=\partial_t+\vec u\cdot\nabla$$ where $$\vec u$$ is the incompressible velocity field (i.e $$\text{div}(\vec u)=0$$. I am to show that for this $$f$$, $$D_t(\vec \omega\cdot\nabla f)=0$$ where $$\vec\omega=\text{curl}(\vec u)$$.

I am able to simplify this using Einstein Summation notation to be $$D_t(\vec\omega\cdot\nabla f)=(\vec\omega\cdot\nabla \vec u)\cdot\nabla f$$ by using the fact that $$D_t\,\vec \omega=\vec\omega\cdot\nabla\vec u-\vec\omega\,\text{div}(\vec u)$$.

What I get hung up on is the step where you must find $$D_t(\nabla f)$$. It seems to me that this should be identical to $$\nabla D_tf$$. Obviously this is incorrect, but I don't understand why...

Thank you for any replies

WnGatRC456
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