What is the difference between dx, Δx and δx?

[Consolacion Ruiz]

What is the difference between dx, Δx and δx?

Δ = difference

d = Δ but small difference, infinitesimal

δ = d but along a curve

Mathematical symbols are always graphics.I’m not sure if that will be true, but it would be beautiful.

 

[Delta2]

We use ##d## for an exact differential https://en.wikipedia.org/wiki/Exact_differential, while ##\delta## for an inexact differential https://en.wikipedia.org/wiki/Inexact_differential.
I am going to highlight some of the things you can read in the above Wikipedia links:

1) In the case of one independent variable x, a differential is a form A(x)dx. If there is a function ##f(x)## such that its derivative is ##f'(x)=A(x)## then the differential is exact and it is written as ##df=f'(x)dx=A(x)dx##.

2) in the case of many variables let's say in the case of 3 independent variables, call them x,y,z a differential is a form ##A(x,y,z)dx+B(x,y,z)dy+C(x,y,z)dz##.

If there is a function ##f(x,y,z)## such that its corresponding partial derivatives with respect to x,y,z equal A,B,C then that differential is called an exact differential and is written a ##df##.

That is ##df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz=A(x,y,z)dx+B(x,y,z)dy+C(x,y,z)dz## for the proper function f (if such f exists) such that ##\frac{\partial f}{\partial x}=A(x,y,z), \frac{\partial f}{\partial y}=B(x,y,z), \frac{\partial f}{\partial z}=C(x,y,z)##

if there is not such a function f then the differential is called an inexact differential and can be written as ##\delta \vec {F}=\vec{F} \cdot d\vec{r}## where F is the vector in ##R^3## with ##\vec{F}=A(x,y,z)\vec{x}+B(x,y,z)\vec{y}+C(x,y,z)dz## and ##d\vec{r}=\vec{x}dx+\vec{y}dy+\vec{z}dz##

 

[Consolacion Ruiz]

Thanks Delta², Your explanation is very clear and a quick response

 

[vanhees71]

The ##\delta x## denotes rather a variation in variational calculus, e.g., in the Lagrange formalism of classical mechanics, where you have an action functional
##A[x]=\int_{t_1}^{t_2} \mathrm{d} t L(x,\dot{x}).##
Then ##\delta x## is a little distortion of a given path. You can define functional derivatives as derivatives of functionals rather in an analogous way as you define partial derivatives of multivariate functions. In the latter case you have independent variables ##x_j## with a discrete index ##j \in \{1,2,\ldots, n \}##, while in the former case you can take ##t## in ##x(t)## (defining a trajectory) as a kind of "continuous index".

 

[Consolacion Ruiz]

Very interesting, then δ is a d with a little distortion Thanks for your explanation.

Δ = Difference

d = Δ, but small difference, infinitesimal

δ = d, but with a little distortion

Mathematical symbols are always graphics.

 

[vanhees71]

##\delta x## is called "variation".