How can the displacement fields be composed to obtain the desired outcome?

[pamparana]

Hello all,

I am reading an engineering book and am having trouble understanding a bit. Let u, v and w be displacement fields, we want w such that

Id+w = (Id + u) o (Id + v)
= Id + v + u_warped_by_v

meaning w = v + u_warped_by_v

In the above equations o denotes composition of the displacement fields.

I am having trouble understanding how

(Id + u) o (Id + v) = Id + v + u_warped_by_v

I thought that by distributive law, we have:

(Id+u)o(Id+v) = (Id+u)o(Id) + (Id+u)o(v)

However, still having trouble trying to derive that expression. Wondering if anyone could help me out with it. Would be much appreciated.

Many thanks,

Luc

 

[HallsofIvy]

It would help if you would explain your terms. I assume that "Id" is the identity operator. I have no idea what "u warped by v" could mean!

If Id is the identity operator, then what you have is NOT correct.

(Id+ u)o (Id+ v)= Id+ u+ v+ u o v. I presume your "u warped by v" is my u o v but it seems to me there must be a "u" in the formula that is not in yours.

 

[pamparana]

Hi,

Thanks for the reply. I read a bit online about function composition and it seems that it is not distributive.

But I still am at a loss as to how this identity came about:

Id+w = (Id + u) o (Id + v) = Id + v + u o v

Thanks,

Luc

 

[tiny-tim]

Hi Luc!

Id+w = (Id + u) o (Id + v) = Id + v + u o v

As HallsofIvy says, that's wrong, it should be Id + v + (u + u o v) …

I guess that "u_warped_by_v" is the book's way of writing (u + u o v)

 

[pamparana]

Hi there,

Thanks for the reply. So what I could come up with is:

(Id + u) o (Id + v) = (Id o (Id + v)) + (u o (Id + v))
= (Id + v) + (u o (Id + v))

However, I am not sure if I can write:

(Id + u) o (Id + v) = (Id o (Id + v)) + (u o (Id + v))

Is that not assuming that composition of group elements is distributive...

Thanks,

Luc