What does this special arrow symbol mean?

[harveyab]

I was reading the Wikipedia article at http://en.wikipedia.org/wiki/Euler-Lagrange_equation and came across several symbols I am not familiar with.
An arrow pointing right with a short vertical bar on the left end. What does this special arrow mean?
How is it different from an arrow without the bar?
Also an arrow with two horizontal lines. What does it mean?
And what is the difference between a regular equal sign "=" and this one ":=", used several times?

Also in a book I was reading there is a symbol of two crossed lines like the letter x or the cross product symbol but quite a bit wider. It occurred twice, but one place it also had two dots above it like a wide umlaut. What are these?

Thanks

 

[hilbert2]

I'm not sure about that arrow, but the symbol ":=" means "is defined to be" unlike regular "=", "equal to".

 

[pwsnafu]

An arrow pointing right with a short vertical bar on the left end. What does this special arrow mean?
How is it different from an arrow without the bar?

"maps to". It's used to define functions. Example the function which takes x to x² would be written as ##x \mapsto x^2##.

Also an arrow with two horizontal lines. What does it mean?

Logical implication.

And what is the difference between a regular equal sign "=" and this one ":=", used several times?

"defined as".

Also in a book I was reading there is a symbol of two crossed lines like the letter x or the cross product symbol but quite a bit wider. It occurred twice, but one place it also had two dots above it like a wide umlaut. What are these?

No idea. Try detexify