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Is jquery a **normal** library to javascript the same way Math is a library to c++, python or even js?

I'm new to web development, I'm discovering javascript and all its capabilities. I'm currently learning jQuery and it made me wonder: "what really is a library ?".
From some so/internet search I understand that jQuery is a library, although some people also consider it a framework as it help/force you to change the way you code js.
For me, a library is a set of functions and classes to help the programmers on a particular point.
As I understand it jQuery doesn't seem to only add function or classes contrary to other javascript libraries such as math, for example or even other jQuery plugins like datatables etc...
To me, jQuery at its base just looks like another way of presenting js code.
So to summarise my questions are:
Is jQuery a "normal" library like Math for c++, python or even js?
How does the js engine understand jQuery? (some sort of compilation ?)
Can every js engine understand jQuery as long as they have the jQuery.js file, or is there something already embedded inside the engine for jQuery.
I also took a look at this very interesting so post on Is a jQuery compiler possible?, but it just blurred me the line between library and jQuery even more.
Any hint or link to how js/jquery combine would be helpful!

Is jQuery a "normal" library like Math for c++, python or even js?
Yes. It's just a collection of functions, etc.
How does the js engine understand jQuery? (some sort of compilation ?)
jQuery is just code written in JavaScript using the standard features of the browser (the DOM, etc.). (Well, and some non-standard ones it feature-tests for to get around browser idiosyncracies.)
Can every js engine understand jQuery as long as they have the jQuery.js file, or is there something already embedded inside the engine for jQuery.
jQuery is designed for the browser environment, so it won't work correctly in a non-browser environment without additional dependencies (a DOM library, etc.).
For the avoidance of doubt: There's nothing you can do with jQuery that you can't do without jQuery, using JavaScript's standard library, the DOM API, and other browser APIs directly. All jQuery does is provide an alternate API for doing DOM manipulation, provide some utility functions, and define a standard way to add features to extend jQuery with more functionality (e.g., "jQuery plugins").
And for completeness:
jQuery is not a language, although it's a common misconception that it is. You don't do something "in jQuery" or "in JavaScript." The correct contrast is "with jQuery" or "with the DOM" (or "without jQuery").
There are other libraries which also fill the same sort of "make the DOM easier" niche, although jQuery is by far the most successful of them.
There are entirely different approaches to handling browser-based interfaces, provided by a spectrum of projects from libraries (like KnockoutJS and many others) through light(ish) frameworks (like React and others) to full-on frameworks (like AngularJS and many others).

Its pretty simple how you understand a library and a framework when it comes to jquery its a library because it dose'nt add any additional jquery approch whatever you see in jquery is present in javascript what jquery does is shorthand your code and help you chain functions which was not that much possible in javascript and when it comes to plugin they all are also part of javascript functions

Ajax and Dojo Ajax

I've been looking into the javascript package Dojo, and I noticed that it uses sort of its own form of Ajax, but as far as I can tell it does the same things as standard Ajax. Is there a benefit to using either over the other, or is there really a difference?
I'm new to both Ajax and Dojo so feel free to correct anything I may have said.

The first line of the Dojo description sums up the benefit: "Dojo saves you time."
You can code all of your AJAX functionality by hand, or you can use a framework (Dojo, jQuery, Prototype, etc) to take care of a lot of the mundane tasks for you.
Edit:
For a good list of reasons to use Dojo or something similar, please see: 6 Reasons To Use JavaScript Libraries & Frameworks.

Like other libraries and frameworks for client-side web applications, Dojo provides a wrapper around the native browser ajax capabilities. Ultimately it can only do what the browsers will allow. It can, however, make things a little more convenient, in a way similar to that by which php makes assembly language programming more convenient.

Javascript: How to make sense of all the frameworks and design philosophies?

I have been a user of jQuery (and some of its minor plugins) for a while. The Javascript code I've developed over the years could be described best as... messy. It used a ton of global variables and functions here and there, didn't use standard ways of organizing the code, nor any design patterns whatsoever.
I am currently building the new version of a website, and I have completed doing the backend with PEAR::MDB2 and Smarty templates. The rest is just homebrew PHP with some classes.
Now I am at the point where I'll add the Javascript layer on top of the website to improve the user-friendliness of some features. (while making sure everything degrades gracefully) I want to write better, cleaner, more organized Javascript than I used to, so I did a little research. I read Stefanov's Object-Oriented Javascript to have a better grasp on some concepts I knew only loosely about (prototypes, constructors, etc.) as well. Now I'm stuck at a point where I wonder which Javascript frameworks I should use, and how to organize it all.
After conducting my research, I understood Cappuccino & Objective-J, and Sproutcore were not what I was looking for. To quote Cappucino's about page:
Cappuccino is not designed for building web sites, or making existing sites more "dynamic". We think these goals are too far removed from those of application development to be served well by a single framework. Projects like Prototype and jQuery are excellent at those tasks
So there's that. Then I found out about Coffee Script, which is more of a one-to-one "compiler" and wouldn't help me with the actual organization of my code.
I also stumbled on some articles that give guidelines:
Using Inheritance Patterns to Organize Large jQuery Applications
A JavaScript Module Pattern
I also found out about Backbone.js, Shoestring, JavaScriptMVC, Google Loader, jQuery Tools, jQuery UI. I don't really know what to do of all this... The things I know:
I don't want to invest too much time in learning something too complex, I want to keep things simple and flexible as much as possible (that is why I don't use Symfony on the backend, for example), yet clean and organized.
I want to use jQuery, the question is, what should I use with it? (that is compatible too)
Right now, I'd use jQuery and jQuery Tools and "organize" all that in a simple namespace/object literal with simple properties and methods and also, since the site is localized, I just plan on using the simple vsprintf (as I do on the backend) with key:value pairs loaded from an object literal provided by the backend. JavaScriptMVC seems interesting, but I fear it would bring way too much complexity for a project that is fairly small sized. That is where I need your advice! Thank you very much in advance.

Ok, my attempt at an answer:
There is no 'best' to way to do it. You now know what's there and I think you might have a preference for yourself for what you want. In that case, pick a framework and learn it inside-out. (sorry to burst your bubble, but each framework has a learning curve, some steep, some very easy, but in the end to use it well you have to invest in it. Just do it, you won't be sorry).
You of course have an preference for clean code, so you might take some considerations into account. You also say you have a preference for jQuery, which is fine, but there are some limitations (as also pointed out in the link provided by eskimoblood).
There are some nice lectures / and tutorials with advice on how to structure your code in jQuery:
How to manage large jquery apps
On Large jQuery apps
Essential Javascript and jQuery patterns (free ebook)
Some style guides:
Jquery core UI Styleguide
Google Closure Javascript Style Guide
Tools for checking your code
JSLint
JSHint (a more forgiving/practical fork)
Closure Linter (haven't tried it yet, but intend to)
Standard works (javascript)
Everything by Douglas Crockford
Quirksmode
There might be more.. perhaps more people can contribute, but I also think that you've almost reached the end of what you can learn before getting your hands dirty. Many of these guides are written in a very generic way, but the interesting thing is that javascript is called upon in many specific situations. It might be useful to just post some of the code that you regard as "messy" and we can help you figure out how to do it better. Good luck!

You should watch the video and read the links in this article and then you should ask yourself again if jquery is the right tool. Maybe you will use dojo, that is much better for larger projects or you take a look at backbone and where you can stay with jquery. After all both of them are more "javascriptish" then something like sproutcore, cappuciono or even GWT. And also much easier to understand when you come from jquery.

One framework that is to consider is definitely ReactJS from Facebook. This framework is pretty slick in many ways.
First thing you have to know is that it is a view framework. It can be used server-side to do the pre-rendering of pages, but it really shines on client side. Since it's a view framework, it can be used with backbone or any other "back-front"-end framework.
One of the main point of React is its rapidity. It keeps a virtual DOM in memory and virtualize all the webpages events. So the virtuals event are used to keep events browser agnostics.
The virtual DOM kind of make programming a dynamic site as if you were programming an old static website. You can just shoot the whole HTML to render to the view engine (as if you were "re-rendering" the whole page) and it will manage the DOM operations. It does a diff between the new virtual DOM and the current virtual DOM and only inserts nodes that needs to be inserted. This way you reduce the number of DOM ops and thus increase your render speed by a lot.
A good place to start is this tutorial which shows how to use "Flux" (the web flow designed by Facebook for its site) in order to realize a Todo application!

Wrap jQuery, dojo with custom library?

Coming from Java, I'm wondering if a Java best practice applies to JavaScript.
In Java, there's a separation of interface and implementation, and mixing them up is considered a bad practice. By the same token, it is recommended to hide implementation details of your library from end developers.
For example, log4J is one of the most popular logging libraries out there but it is recommended to write code to the slf4j library or the Commons Logging library that "wraps" log4j. This way, if you choose to switch to another logging framework such as logback, you can do so without changing your code. Another reason is that you, as a user of a logging library, how logging is done is none of your concern, as long as you know what logging does.
So back to JavaScript, most non-trivial web applications have their own custom JavaScript libraries, many of which use open source libraries such as jQuery and dojo. If a custom library depends on, say jQuery, not as an extension, but as implementation, do you see the need to add another layer that wraps jQuery and makes it transparent to the rest of JavaScript code?
For example, if you have the foo library that contains all your custom, front-end logic, you'd introduce the bar library that just wraps jQuery. This way, your foo library would use the bar library for jQuery functions, but it is totally oblivious to jQuery. In theory, you could switch to other libraries such as dojo and google web toolkit without having a big impact on the foo library.
Do you see any practical value in this? Overkill?

Although it makes sense from a theoretical standpoint, in practice I'd say it's overkill. If nothing else for these two reasons:
Anything that adds to the size of
the request (or adds more requests)
is bad - in web world, less is more.
If you're using say jQuery, the
chances of you switching to
something like Mootools is (imho) slim to none. From what I've seen, the top libraries each aim to solve different problems (at least in the case of Mootools and jQuery - see this great doc for more info on that). I'd assume that you'd incur a tremendous amount of headache if you were to try to implement a middleware library that could easily switch between the two.

In my experience and being a Java developer myself, sometimes we tend to take the whole "abstraction" layer pattern too far, I've seen implementations where someone decided to completely abstract a certain framework just for the sake of "flexibility" but it ends up making things more complicated and creating more code to maintain.
Bottom line is you should look at it on a case by case basis, for example you wouldn't try to create an abstraction layer on top of struts, or on top of JPA, just in case you then go to a different framework (which I've rarely seen done).
My suggestion is, regardless of the framework you are using, create objects and components that use the framework internally, they should model your problem and be able to interact between them without the need of any specific framework.
Hope this helps.

There are a lot of good answers here, but one thing I don't see mentioned is feature sets. If you try to write a library to wrap the functionality provided by, say, jQuery, but you want to be able to easily swap out for something like prototype, you have a problem. The jQuery library doesn't provide all the features prototype provides, and prototype doesn't provide all the features jQuery provides. On top of that, they both provide their features in radically different ways (prototype extends base objects -- that's damn near impossible to wrap).
In the end, if you tried to wrap these libraries in some code that adds 'abstraction' to try to make them more flexible, you're going to lose 80% of what the frameworks provided. You'll lose the fancy interfaces they provide (jQuery provides an awesome $('selector') function, prototype extends base objects), and you'll also have to decide if you want to leave out features. If a given feature is not provided by both frameworks, you have to either ditch it or reimplement it for the other framework. This is a big can of worms.
The whole problem stems from the fact that Java is a very inflexible language. A library provides functionality, and that's it. In JavaScript, the language itself is insanely flexible, and lets you do lots of crazy things (like writing a library, and assigning it to the $ variable). The ability to do crazy things lets developers of javascript libraries provide some really creative functionality, but it means you can't just find commonalities in libraries and write an abstraction. I think writing javascript well requires a significant change in perspective for a Java developer.

Someone wise once said "premature optimization is the root of all evil." I believe that applies in this case.
As others have expressed, you don't want to abstract for the sake of flexibility until you have an actual need for the abstraction. Otherwise you end up doing more work than necessary, and introducing unnecessary complexity before it is required. This costs money and actually makes your code more brittle.
Also, if your code is well organized and well tested, you should not be afraid of major changes. Code is always changing, and trying to anticipate and optimize for a change that may or may not come will almost always get you in more trouble than it saves you.
Acknowledgement: I should give credit to Agile programming and my practice and readings on the topic. What I've said comes directly from my understanding of Agile, and I've found it to be an extremely good razor to cut out the extra fat of my work and get lots done. Also none of what I've said is actually JavaScript specific... I'd apply those principles in any language.

There are good arguments calling this development practice - wrapping in order to switch later - into question in any language.
A good quote by Oren Eini, from his writeup on wrapping ORMs:
Trying to encapsulate to make things
easier to work with, great. Trying to
encapsulate so that you can switch
OR/Ms? Won’t work, will be costly and
painful.

This is definitely something that is done in enterprise environments.
Take for example a company that has their own custom javascript framework that is used on all of their projects. Each of the projects decide to use their own framework (jQuery, Dojo, Prototype) to add functionality to the underlying modules of the company framework. Employees that move between projects can now easily do so because their API with working the project's codebase is still the same, even though the underlying implementation could be different for each project. Abstraction is helpful in these situations.

It is overkill. Javascript is not Java and is not in any way related to Java. It is a completely different language that got J-a-v-a in the name for marketing reasons.
If you are concerned with availability of add-on libraries, then choose a framework with a large ecosystem. In an enterprise environment you will be further ahead by standardising on a vanilla off-the-shelf uncustomised web framework that you can upgrade every year or so tracking the rest of the world. And then supplement that with a SMALL in-house add-on library which you will, of course, have to maintain yourself, not to mention training any new programmers that you hire.
Since you are talking about Javascript in the client (web browser) it is more important that you limit the complexity of the things that people do with it. Don't build huge amounts of client side code, and don't make stuff that is so brittle that another programmer can't maintain it. A web framework helps you both keep the linecount down, and keep your own code reasonably simple.
It is not a question of Javascript best practice, because that would be different for server-side JS such as Rhino or node.js.

Adapter pattern is not a common solution in this case. The only example I know to use this pattern is extjs. Javascript projects are usually too small and they aren't worth the effort you would make by creating such an abstraction layer.
The common solution for this problem is that you try to use multiple frameworks together for example with jquery.noConflict.

I've done this before, and can talk a bit about the experience of writing a library/toolkit wrapper.
The plan was to move from Prototype to some other library. Dojo was the first choice, but at the time I wasn't sure whether that's the library to move everything to (and by everything I mean ~5MB of Prototype-happy JS). So coming from a world of clean interfaces, I was set to write one around Prototype and Dojo; an awesome interface that would make switching out from dojo a breeze, if that was in fact necessary.
That was a mistake that cost a lot of time and effort for a few reasons. The first one is that although two libraries can provide the same functionality, (a) their API will almost always be different, and most importantly (b) the way you program with one library will be different.
To demonstrate, let's take something as common as adding a class-name:
// Prototype
$("target").addClassName('highlighted');
// Dojo
dojo.addClass("target", "highlighted");
// jQuery
$("target").addClass("highlighted");
// MooTools
$('target').set('class', 'highlighted');
Pretty straight-forward so far. Let's complicate it a bit:
// Prototype
Element.addClassName('target', 'highlighted selected');
// Dojo
dojo.addClass("target", ["highlighted", "selected"]);
// jQuery
$("target").addClass(function() {
return 'highlighted selected';
});
// MooTools
$("target").set({
"class": "highlighted selected"
});
Now after choosing an interface for your version of the addClass you have two options: (1) code to the lowest common denominator, or (2) implement all of the non-intersecting features of the libraries.
If you go with the 1st -- you'll loose the "personality" / best qualities of each of the library. If you go with #2 -- your addClass' code will be at 4 times larger than the ones provided by any of the libraries, since for example when Dojo is included, you'll have to write the code for the function as the first param (jQuery) and the Object as the first param (MooTools).
Therefore, although it is theoretically possible, it isn't practical, but is a very nice way to understand the intricacies of the libraries out there.

jQuery and Plugin Namespacing and Inheritance

Hey guys. There has been a lot of activity lately on the jQuery Dev Group about prototypal inheritance and plugin namespacing, and I want to see who has the best answer for it.
Group link:
http://groups.google.com/group/jquery-dev/browse_thread/thread/620c6a18a16d9665
Questions:
What do you guys think should be done about this and why?
Do you agree with any opinions there? Why or why not?
Why it is important:
If jQuery did decide to implement something like this into the core, this is a huge decision that will affect a lot of people and may affect how they use/extend jQuery and if they continue to use jQuery. It may convert a lot of people to jQuery.

I think what sets jQuery apart from libraries like prototype is that it concentrates on features such as fast DOM selection, traversal and manipulation rather than adding 'flavor' to javascript.
Problem with Prototype is it puts a ruby hat on javascript and it ends up getting bloated with features you probably never end up using.
I have worked in projects where people overuse prototype Classes to implement everything as a class where it simply wasn't needed. People who move from Java or C# (or even Ruby) like to do everything their way - which is often not needed.

I think that this is important step for jQuery. Of cause jQuery is very dom centric lbrary. And that is why it achieves so much attention over the world, but it is hard to build complex solutions basing on it only. You can not use all the benefits of OOP and so you must to have all the code in a head - all of your programmers must. I suppose jQuery is the most easy solution - thats why everybody likes it, but now it wants to be enterprise solution.
I think this can make enterprise client side development faster and that is great:)