Can saomebody explain me whats differential mean?

[Mzz]

Can saomebody explain me what's differential mean??

can saomebody explain me what's differential mean??

 

[Kummer]

Go to Wikipedia at www.wikipedia.com and type "Differential".

 

[Mzz]

ı know that.. thanks

 

[cristo]

It depends upon what context you are using the word differential in. Could you describe what you're studying?

 

[HallsofIvy]

Well, while "differentials" are used in "differential equations", you need to go back to calculus to find a definition. I'm moving this thread to "Calculus and Analysis".

 

[T.Engineer]

can saomebody explain me what's differential mean??

let Δx = dx be an increment given to x.
then
Δy = f (Χ +ΔΧ) - f(Χ)
is called the increment in y=f (x). if f(x) is continuous and has a continuous first derivative in an interval, then
Δy = f ' (Χ) ΔΧ + ε ΔΧ = f'(x) dx + ε dx

where ε goes to 0 as ΔΧ goes to 0.
the expression
dy = f' (x) dx.
where dy is called the differential of y

 

[WWGD]

A way of seeing a differential that makes sense to me is this:

The differential measures the change of the function along the

tangent line, plane (depending on wether you are in R,R^2, etc.)


A function is differentiable , if the change in its values can be

approximated locally by a linear function, to any degree of precision

( in a delta-epsilon sense). The linear function that does this approximation

is itself the derivative.

This is one way of seeing the definition:


|| f(x+h)-f(x) -L(h)||=0
Limh_>0 __________
h

An example:

For f(x)=x^2, we have:

df=f'(x)dx , so df=2xdx.


This means that the change in the value of f(x)=x^2 in

a 'hood ( 'hood = neighborhood) of a point can be approximated

by the change in the values in 2x.


Take a small 'hood of, say, 10 on the real line, take

(9.9,10.1). The change of f(x)=x^2 from the value 10 is:


i) |10.1^2 -10^2| =0.201

ii) |9.9^2-10^2| =0.199


Now, consider the approximation to the change of x^2 ,using the derivative:

i') |2(10.1)-2(10)|= 0.2

ii') |2(9.9)-2(10)| = 0.2


The error is pretty small, right?. There are, of course, analogies to this
in higher dimensions, with approximations along tangent planes, etc.
Unfortunately things get much hairier outside of R^n, where you have
sometimes just local Euclidean, like in manifolds, without the standard
tangent planes.


Hope that helped

 

[mathwonk]

a curve is tyhe graph of a non linear function. its family of tangent lines are translates of the graphs of a family of linear approximations to this functions, and this family of linear functions is called the differential of the function. at each point, the tangent line is the graph of the differential at that point. It has equation f'(a)(x-a), at a. (A linear approximation to f(x)-f(a).)

 

[WWGD]

Nicely written!. Sorry I could not find a clearer way of writing it. Do
you think my post was overall correct (tho, I admit, not too clear
nor to the point.).

I had a follow-up question, please:

Does the Jacobian J(f) map a point to its differential?. If so, what
is the relevance of the differential being 0 at a point?.

Thanks.