User talk:Laniakeani

/* Please explain under heading special notations: (A dot NABLA) in brackets... */ new section
I WAS unfamiliar with this use of the Nabla symbol, it seems out of order.. I thought it was a typo. BUT it appears more than once (its also part of Green's vector identity), the heading: cross product rule, and Vector dot Del Operator... ALSO and Strangely it appears in an expression under summaries, under gradient identities... its the last identity ... and it is presented to us before presenting the summaries for the dot Del Operator..

I had to dig other resources to identify its meaning as the 'adjective or convective operator' https://www.physicsforums.com/threads/meaning-of-a-dot-nabla-b.589069/

I am not a subject matter expert on this but I believe the above link has correct information WRT my question, and I would like to see my question answered under special notations... I would be more than happy to make the edit if other people agree that the physicsforums.com link is correct!

Thanks for your attention! Steve

PS. I think the Vector dot Del Operator heading is really something to do with a special notation that is not fully explained. I got the best understanding here:

https://www.quora.com/What-is-the-difference-between-%E2%88%87-A-and-A-%E2%88%87 where it says:

"To perform dot product A must be a vector field. For both the cases use general rule of dot product. But del is an spatial differential operator. Although dot product is commutative. But here it is not so. ∇.A represents a physical quantity called divergence. Whereas A.∇ will give another operator (Spatial differential) which can operate on another function to give certain results. But of course the result will not contain any significant meaning and of no use. So for useful purpose in classical Physics we will have to use ∇.A and need not to bother about A.∇ . Hope it helps.

But yes in quantum mechanics A.∇ is important in operator algebra."

PSS... Now I think I understand it better after studying more.. I believe its a general concept for a DIRECTIONAL 'spacial differential' (referencing term used above), basically its a spacial differential with a vector bias..and it appears in what is called the directional derivative... https://en.wikipedia.org/wiki/Del#Directional_derivative

PSSS...

https://www.physicsforums.com/threads/a-dot-del.157380/post-1251785 Confusion abounds about interpretation: there is an 'abuse of notation' interpretation similar to the divergence (because of NABLA in this case needs plus signs in it to make a scalar).... resulting in an interpretation that is essentially the divergence with a vector directional bias still resulting in a scalar result. Then there is a non-abusive interpretation where it is a vector similar to NABLA-without-plus-signs but with a directional bias. So is it scalar or is it vector? If its a vector, how does it differ from the gradient? If its scalar how does it differ than the divergence? And what convenience does it bring to operations that we are familiar with already? BTW because of abuse-of-notation, there is no independent interpretation of NABLA.... NABLA needs to be combined with some other symbol to make sense, and that other symbol needs to be put very close to it.... this resolves the 'abuse of notation' in a pragmatic way... but only by agreed upon convention, which I am still trying to understand... and I understand now, why most recently, some math ISO committee dropped the NABLA convention entirely and are now using function references... eg curl, grad, div.... there is also my new schooling of exterior differential calculus which is more precise....nevertheless my quest goes on...

PSSSS... so many derivatives so little time it seems... and the name space is so cluttered .... adjective and convective seem to refer to same thing and the name space gets more cluttered now that I found the term 'material derivative' https://en.wikipedia.org/wiki/Material_derivative... I found these names all refer to the same thing (or perhaps they are slight variations on the same formula) and this same formula is a special case of the 'total derivative' https://en.wikipedia.org/wiki/Total_derivative "Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously." I will meditate on this to try and discover the not many situations... the abuse of learning an old-fashioned interpretation goes on....

PSSSSS.... I am finding an consistent interpretation... as the material derivative for Euclidean Co-ordinate system... https://en.wikipedia.org/wiki/Material_derivative#Orthogonal_coordinates...manipulating two variables, it produces a column vector instead of a column matrix, A ( the vector on the right) is a total vector in the sense that each component of A is differentiated by all the components possible in the spacial vector, while u (the vector the left that is dotted with nabla) acts as a component-wise multiplier for the spacial differential. so we have a component-wise spacial differential component multiplier, that is being dotted, at the same time we got total spacial differential whereby each additive spacial differential in each final vector component is dedicated to component-wise to one component of A. The setup is general enough thus resulting in a vector if the variables are n-vectors, or a scalar if they are both scalars. Phew! So there is both a scalar and vector interpretation! I am also finding an consistent interpretation here.... https://mathworld.wolfram.com/ConvectiveDerivative.html

major typos in ref to BOTH Maxwell equations... is there a way to link equations to avoid making miscopy typos?
https://en.wikipedia.org/wiki/Stokes%27_theorem#Maxwell's_equations

Wait! this is not how I remember Maxwell equations... but it might be just a sub-component of the usual way Faradays and Ampere laws are expressed... what I see is consistent with regard to Stokes Law.