Recent content by SamJohannes

So I've done a heap more. But now I've come up against some more tricky ones. They are i)$\nabla(u\cdot v)=(u\cdot\nabla)v+(v\cdot\nabla)u+u\times(\nabla\times v)+v\times (\nabla \times u)$ and ii)$u\times (\nabla\times u) = \frac{1}{2}\nabla (u\cdot u) - (u\cdot\nabla )u$ For number one I...

Ah, I see. So I got: $\nabla\cdot\left(\phi\textbf{u}\right)=\partial_{i}\left(\phi{u}_{i}\right)=\phi\left(\partial_{i}{u}_{i}\right)+{u}_{i}\left(\partial_{i}\phi\right)=\phi\left(\nabla\cdot\textbf{u}\right)+\textbf{u}\cdot\left(\nabla\phi\right)$

Hi Everyone! I'm looking to prove $\nabla\cdot\left(\phi\textbf{u}\right)=\phi\nabla\cdot\textbf{u} + \textbf{u}\cdot\nabla\phi$ in index notation where u is a vector and phi is a scalar field. I'm unsure how to represent phi in index notation. For instance, is the first line like...

Double check this one. You've left out the side lengths from your derivative

I think there's a mistake in your expanding. You must remember to square the entire right side of the equation, not the individual terms...

Hello hossam killua! Here's how I'd approach it. First I'd expand the partial derivatives $\pd{z}{u}$ and $\pd{z}{v}$ using the chain rule. From there you can find ${\left(\pd{z}{v}\right)}^{2}$ and ${\left(\pd{z}{u}\right)}^{2}$. When you add them together and sub in the known partials you...

Sorry about any lack of clarity. Since the start of this post I learned a little bit about Latex so here's my attempt. If got to here, which is what I had at that beginning but I poorly explained it. $$\varepsilon_{ijk}\partial_i\partial_ju_k=u_k\varepsilon_{kij}\partial_i\partial_j$$ I guess...

Thanks Ackback! You showed clear steps and I understand the rest just cancels out via Clairaut's thm. I didn't make it clear that the question asks for the solution in tensor notation. So is this a perfectly viable way to prove it in this manner? Cheers

Hi everyone! I've got a vector index notation proof that I'm struggling with. (sorry ignore the c, that's the question number) I've simplified it u * (del X del) and from there I've sort of assumed del X del = 0. Is that right and if so could somebody please explain it? Else any help on...

Any thoughts on part b?

Thanks Opalg. That's helped a lot.

Thanks for the response Opalg, it's good to be here. You're right, ∥x∥1 is the 1-norm. I don't understand the bit ∥x∥=∥∥∑xjej∥∥. Is this true for all norms? Sorry if the question sounds silly, I'm relatively new to the topic. -Cheers, Sam

Hi guys, I've attached a problem that I've been struggling with for a while now. I was wondering if anyone had some advice on how to approach it (in particular part a) or some resources they could recommend to me?Thanks in advance, Sam