No. If ##u=x+1## then ##x=u-1## and you get ##f(x)=f(u-1)=\sqrt{u}=\sqrt{x+1}\,.##Hi.
If I have a function f(x) = √(x+1) and I define u=x+1 is it correct to state f(u) = √u ?
##f(u)=\sqrt{u+1}##. The name of the variable doesn't matter. You could as well write ##f(tree)=\sqrt{tree+1}##, but this would be a bit confusing and too long to use.Thanks , what would f(u) be then ?
Again, no. ##f(u) = \sqrt{u + 1}##, as already explained by @fresh_42.Consider f(z) = √ (z+1) , now define u = z+1 then f(u) = √u ,
This part isn't relevant to what you asked about.dyn said:this is then used to show that f(u) is multi-valued around u=0 and so f(z) is multi-valued around z= -1
A number of physics text books will use this shorthand but it is mathematically imprecise. They also skate over the chain rule somewhat. What you are really trying to do is to find a new function ##g## such that:Hi.
If I have a function f(x) = √(x+1) and I define u=x+1 is it correct to state f(u) = √u ?
This is an abuse of notation.No , there was no integral. Another similar example is differentiation using the chain rule , for example if
##y(x) = (x+1)^2## then using u=x+1 , ##y(u) = u^2## and then this is differentiated using the chain rule
I do not think so. As you see in the picture, there are two different curves. Now if you only consider their slopes, then they behave the same (at different points). If you integrate them, then the results will be the same (within different integration limits). In neither case it is a matter of notation. They were and will be two different functions. This is especially important for physicists, as the frames are important here! A mathematician could say "I don't care, I'm only interested in the geometric object", but a physicist cannot. Change of coordinates is completely uninteresting for mathematicians, physicists do nothing else! So it is not an abuse of notation, it is a lack of description from your part. Your initial question is a strict NO, and then you came up with the chain rule, which didn't make any sense without further context. ##f(u)=\sqrt{u+1}## if ##u=x## and ##f(u-1)=\sqrt{u}## if ##u=x+1##. There is literally nothing which can be abused! ##g(u)=\sqrt{u}## is a different function, in physics as in mathematics. Fullstop.Thanks for all your replies. I think the problem is the difference between the way physicists and mathematicians are strict/not strict about notation
I was quite happy with the answers before this post but the above post does not make any senseI do not think so. As you see in the picture, there are two different curves. Now if you only consider their slopes, then they behave the same (at different points). If you integrate them, then the results will be the same (within different integration limits). In neither case it is a matter of notation. They were and will be two different functions. This is especially important for physicists, as the frames are important here! A mathematician could say "I don't care, I'm only interested in the geometric object", but a physicist cannot. Change of coordinates is completely uninteresting for mathematicians, physicists do nothing else! So it is not an abuse of notation, it is a lack of description from your part. Your initial question is a strict NO, and then you came up with the chain rule, which didn't make any sense without further context. ##f(u)=\sqrt{u+1}## if ##u=x## and ##f(u-1)=\sqrt{u}## if ##u=x+1##. There is literally nothing which can be abused! ##g(u)=\sqrt{u}## is a different function, in physics as in mathematics. Fullstop.
All what came after post #2 (or #4) is pure guesswork (including mine, with the exception of this one) based on lacking context, information and clarity.