Vector Differentiation - Suffix Notation.

Having a bit of trouble comprehending the following expression in suffix notation:

F_j (∂/∂x_i)G_j

Now it's a sum over j so its

F₁(∂/∂x_i)G₁ + F₂(∂/∂x_i)G₂ + F₃(∂/∂x_i)G₃

= (F₁, F₂, F₃).((∂/∂x_i)G₁, (∂/∂x_i)G₂, (∂/∂x_i)G₃)

= F . ((∂/∂x_i)G₁, (∂/∂x_i)G₂, (∂/∂x_i)G₃)

What is the second bracket? Is it div(G) (i.e factoring out the (∂/∂x_i) gives (∂/∂x_i)G )

Or is it (grad G₁, grad G₂, grad G₃)?

Reply 1

Cexy

The monster you have there is something called a tensor - but don't worry about that because you don't need to know it yet. The key to understanding it is to treat the bracket with the Gs like a 3x3 matrix where the ij^th element is

\frac{\partial G_j}{\partial x_i}

, and treat F as a vector, and just multiply them out in the appropriate manner. This is what you'd expect - the expression you start with has one free index, so it'll look like a vector.

I think that what's confusing you is that there isn't a common representation for this particular combination. It's not grad or div or anything like that - it's just somethign on its own. There isn't really a nice way to think about it. It took me ages to get to grips with some of the dodgy constructions you get with suffix notation!

Reply 2

Dan R

Cexy

I think that what's confusing you is that there isn't a common representation for this particular combination. It's not grad or div or anything like that - it's just somethign on its own.

Yeah, exactly! I'm still struggling to recognise things straight away in terms of suffix notation, so I want to be able to think about what an expression means in terms of recognizable notation... But if there isn't one, that's fine. Thank's a lot for clearing that up! :smile: