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How to create and implement a custom JavaScript library

I am new to javaScript and am unsure how to go about creating a new js library and referencing this in another js file.
If I have a standalone file Utilities.js
var Utilities=
{
show: function(input)
{
alert(input);
}
};
Am I missing something as to how a library should be defined standalone?
My second question is how to use that is sub-sequent js files. What I did so far is:
<script type="text/javascript" src="resources/Utilities.js"></script>
In my index.html. Is this enough to reference it as:
Utilities.show("Hello");
In any other java script file?
I tested it in this fashion and got and error "Utilities is not defined"
NOTE: This is just an example and not my full and practical purpose.

Yes, including that Javascript file with that global variable declared is enough to call your methods this way Utilities.show("Hello"); from another Javascript file loaded after Utilities.js or inside a <script></script> section of your html.
But you can actually improve it a little, following the module pattern and exposing only the functions you really need to the global scope (you'll likely write some functions that the users of your library should not call directly, this allows you to do it in a clean way):
var Utilities=Utilities||(function () {
//Accessible only here
var privateArray=[];
//Cannot be called from outside this function
var privateFunction=function(){
}
//Return only what must be publicly accessible, in this
//case only the show() method
return {
show: function(input){
privateFunction();
alert(input);
}
}
})();
That (function () { /* Code */})();, defining a new scope for your code will also avoid name clashes with other global javascript object/functions.

It is OK to use object literals, but you can define libraries using other patterns (module, revealing module, constructors, etc).
I recommend these links to understand primitives, scopes, closures, object literals, etc.
http://bonsaiden.github.io/JavaScript-Garden/
http://jsbooks.revolunet.com/
To call the method inside index.html you need to add a tag.
<script>
Utilities.show("Hello");
</script>
But this approach it's not recommended. Instead of it, you can create a new JS file to run your library code.
<script type="text/javascript" src="resources/Utilities.js"></script>
<script type="text/javascript" src="resources/main.js"></script>
In main.js
Utilities.show("Hello");
Hope it helps.

Given the fact that you gave, within yout question, zero context of what you're trying to achieve, the best answer to your original question is that it depends.
If you just need a bunch of files and you're done (like in your example, Utilities.js and a few more) then you're ok with the way you're heading to.
But of course, you'll allways want to scale your front end and thus you should adhere to some architectural pattern. So, if you're building a client side (browser-side) application, then you should really implement your libraries using the module pattern, and begin your project from a good project example / scaffold.
On the other hand, if you're rendering the html on server (e.g. you're using PHP to render the final html file that will be sent to browser) and you just need some thin functionality in the browser, the way you begun can be okay if you're careful. Also, you can still implement the module pattern here too, although I strongly suggest that you should make use of namespacing to have a clear separation of concerns.

In browser based javascript you can't just call functions from different files yet. In Es6 there are ways. But not yet. Which mean just because you have some variable or function etc then you cant reference it automatically in another file.
Unless both files are loaded into one html and are loaded in order.
Alternatively you could run task runner like grunt and 'merge' them upon each build.
Javascript doesnt have special concept of library, in es6 it's a little different, everything is an object.
What you are doing is just creating an object. and yes it will work.

Why does requirejs allow you to specify a string id?

In requirejs it is possible to define an anonymous module or to give it a string id. According to this article, you would normally not use the string id:
You would not normally use the id when you define your module. It is typically used by tools when optimizing a RequireJS application.
I currently define my modules anonymously and use require.config.paths for the mappings. What I don't understand is: why does requirejs allow you to specify string id's if they're not needed?

why does requirejs allow you to specify string id's if they're not needed?
They only are not needed if require can figure out what module it is that just called define. This is the standard when require() did load the script file that contains the module, whose name and path it knows.
However, the optimizer will put multiple modules in a single file, and there needs to be a different way to figure out what modules are define()d. From the docs:
These [names] are normally generated by the optimization tool. You can
explicitly name modules yourself, but it makes the modules less
portable -- if you move the file to another directory you will need to
change the name. It is normally best to avoid coding in a name for the
module and just let the optimization tool burn in the module names.
The optimization tool needs to add the names so that more than one
module can be bundled in a file, to allow for faster loading in the
browser.

I can't answer as to James Burke's motivations, but I can point out instances of its usefulness.
Define your own single page "layer" for testing, using JSBin or JSFiddle. The following code can be easily executed without having to stand up endpoints for each module or to use r.js to create a layer.
define('A',[], function(){ console.log('A loaded');});
define('B',[], function(){ console.log('B loaded');});
define('c',['A','B'], function(){ console.log('C loaded');});
Define a "local override" for troubleshooting. Add a define right before your require to easily preempt a module definition add a new module so you don't have to touch several files while you're working
define('plugin/fancySelect',[], function(){/* ... */});
require([ /* ... */], function(){
// your main application code
});

Using RequireJS with legacy code

I'm working with a very large project that uses:
Legacy JSP pages that includes javascript files with script tags
Backbone models and views that uses other javascript modules without RequireJS
We now want to start using RequireJS with jQuery, BackboneJS and UnderscoreJS for everything we develop from now on, but we don't have the resources to rewrite all the legacy JSP pages. We may have time to rewrite the Backbone models and views we have already developed.
The problem is that for our legacy code (both 1 and 2 above) we include all our javascript files in a huge file and ship to the browsers. This huge file must be able to co-exist with our new RequireJS templates, but how can I e.g. separate certain parts of the huge file, so I can use them with the templates using RequireJS? Without having to rewrite all pages that uses this part of the file, or having duplicate code.
Thanks!

I don't know if I fully grasp the problem at hand, but I think a the shim or map functions of RequireJS will help you out.
Extract the parts you want in a new module from your huge javascript file. Then tell RequireJS that your huge javascript file is a dependecy for this new module using shim. Something like:
requirejs.config({
shim: {
'new-module': {
deps: ['huge-javascript-file'],
exports: 'NewModule'
}
});
Shim documentation: http://requirejs.org/docs/api.html#config-shim
The map function might be useful when only portions of your new code have to use your old huge file. Check out this documentation: http://requirejs.org/docs/api.html#config-map

I don't think there is One True Way to achieve this, but I've approached a similar problem by defining module "facades" around the globally scoped code. Let's say your legacy scripts define a global variable called foo. You can define a AMD module and export that variable from it:
//foo.js
define(function() {
return window.foo;
});
//bar.js
define(['foo'], function(foo) {
//do something with foo
});
This way you only need to write a single-line facade every time you need to use a new piece of the existing, globally defined code, without breaking existing code which expects the same code to be globally defined. Over time you can move and refactor the actual implementation into the module without breaking consumer code.

Questions about Dojo modules from Dojo newb?

I'm new to Dojo so I don't quite understand all of Dojo's features. Here are some questions, but I'm sure some of them seem really stupid since I don't quite get how Dojo is structured yet:
How do you create multiple modules within a single js file and access the module in the file it's created? Also, how do you access specific modules from a file containing multiple modules?
What is the difference between require and define?
I successfully required a module from a file, but I can't figure out how to get the variables out of the file, how do you do this?
I was looking at how Dojo required its modules and notice that it does a http request for each file, but isn't that really inefficient when you're dealing with lots of modules and/or on a large site, you really would like to minimize the number of http requests necessary? What's the way around this?

Reading through The Dojo Loader will provide answers.
Basically module = file and very often (as a matter of best practice) module = file = class (more precisely public class defined via dojo/_base/declare).
Ad #4: You will need to employ The Dojo Build System, that will resolve all the dependencies and put all your modules into a single file (or more files, this depends on your build profile). Have a look at Dojo Boilerplate project, it may help with building your application.

How do you create multiple modules within a single js file?
While this is possible to do, and is done by the build system when it creates layer files, it is not recommended. Putting multiple modules in a single file means you have to use a different form of define that gives an explicit ID to eahc of them. This is less flexible then having the module IDs automatically derived from the file name and path (this, together with relative paths makes it much easier to move and rename AMD modules, compared to old-style modules)
What is the difference between require and define?
define is a beefed up version of require that defines a new module with its return value.
I successfully required a module from a file, but I can't figure out how to get the variables out of the file.
I am not sure wha toyu mean with this (without giving a concrete runnable example) but all you have to do is create an object to be the value of your module and return it from define. This is not very far off from how you would define modules the old way, with manual "namespaces"
//foo.js
define([], function(){
var M = {}; //the exported stuff will go here
var myVariable = 16; //private function variables are private (like they are everywhere else)
//and cannot be seen from the outside
M.anumber = myvariable + 1; //things on the returned object can be seen from the outside
return M; //people who require this module will get this return value
});
//bar.js
define(['foo'], function(foo){
console.log( foo.anumber );
});
I was looking at how Dojo required its modules and notice that it does a http request for each file...
As phusick pointed out, the build system can be used to bundle all the modules into a single file (giving you the best of both worlds - modularity during development and performance during deployment). It can also do other cool stuff, like bundling CSS and passing the Javascript through a minifier or through the Closure compiler, checking ifdef for creating platform-dependent builds, etc.

Javascript object dependencies

In complex client side projects, the number of Javascript files can get very large. However, for performance reasons it's good to concatenate these files, and compress the resulting file for sending over the wire. I am having problems in concatenating these as the dependencies are included after they are needed in some cases.
For instance, there are 2 files:
/modules/Module.js <requires Core.js>
/modules/core/Core.js
The directories are recursively traversed, and Module.js gets included before Core.js, which causes errors. This is just a simple example where dependencies could span across directories, and there could be other complex cases. There are no circular dependencies though.
The Javascript structure I follow is similar to Java packages, where each file defines a single Object (I'm using MooTools, but that's irrelevant). The structure of each javascript file and the dependencies is always consistent:
Module.js
var Module = new Class({
Implements: Core,
...
});
Core.js
var Core = new Class({
...
});
What practices do you usually follow to handle dependencies in projects where the number of Javascript files is huge, and there are inter-file dependencies?

Using directories is clever, however, I think you might run into problems when you have multiple dependencies. I found that I had to create my own solution to handle this. So, I created a dependency management tool that is worth checking out. (Pyramid Dependency Manager documentation)
It does some important things other javascript dependency managers don't do, mainly
Handles other files (including inserting html for views...yes, you can separate your views during development)
Combines the files for you in javascript when you are ready for release (no need to install external tools)
Has a generic include for all html pages. You only have to update one file when a dependency gets added, removed, renamed, etc
Some sample code to show how it works during development.
File: dependencyLoader.js
//Set up file dependencies
Pyramid.newDependency({
name: 'standard',
files: [
'standardResources/jquery.1.6.1.min.js'
]
});
Pyramid.newDependency({
name:'lookAndFeel',
files: [
'styles.css',
'customStyles.css'
]
});
Pyramid.newDependency({
name:'main',
files: [
'createNamespace.js',
'views/buttonView.view', //contains just html code for a jquery.tmpl template
'models/person.js',
'init.js'
],
dependencies: ['standard','lookAndFeel']
});
Html Files
<head>
<script src="standardResources/pyramid-1.0.1.js"></script>
<script src="dependencyLoader.js"></script>
<script type="text/javascript">
Pyramid.load('main');
</script>
</head>

This may be crude, but what I do is keep my separate script fragments in separate files. My project is such that I'm willing to have all my Javascript available for every page (because, after all, it'll be cached, and I'm not noticing performance problems from the parse step). Therefore, at build time, my Ant script runs Freemarker via a little custom Ant task. That tasks roots around the source tree and gathers up all the separate Javascript source files into a group of Maps. There are a few different kinds of sources (jQuery extensions, some page-load operations, so general utilities, and so on), so the task groups those different kinds together (getting its hints as to what's what from the script source directory structure.
Once it's built the Maps, it feeds those into Freemarker. There's a single global template, and via Freemarker all the script fragments are packed into that one file. Then that goes through YUI compressor, and bingo! each page just grabs that one script, and once it's cached there's no more script fetchery over my entire site.
Dependencies, you ask? Well, that Ant task orders my source files by name as it builds those maps, so where I need to ensure definition-use ordering I just prefix the files with numeric codes. (At some point I'm going to spiff it up so that the source files can keep their ordering info, or maybe even explicitly declared dependencies, inside the source in comment blocks or something. I'm not too motivated because though it's a little ugly it really doesn't bother anybody that much.)

There is a very crude dependency finder that I've written based on which I am doing the concatenation. Turns out the fact that its using MooTools is not so irrelevant after all. The solution works great because it does not require maintaining dependency information separately, since it's available within the javascript files itself meaning I can be super lazy.
Since the class and file naming was consistent, class Something will always have the filename Something.js. To find the external dependencies, I'm looking for three things:
does it Implement any other
classes
does it Extend any other
classes
does it instantiate other classes
using the new keyword
A search for the above three patterns in each javascript file gives its dependent classes. After finding the dependent classes, all Javascript files residing in any folder are searched and matched with this class name to figure out where that class is defined. Once the dependencies are found, I build a dependency graph and use the topological sort algorithm to generate the order in which files should be included.

I say just copy and paste this files to a one file in an ordered way. Each file will have a starting and ending comment to distinguish each particular code.
Each time you updated one of the files, you'll need to updated this file. So, this file need to contain only finish libraries, that not going to changes in the near time.

Your directory structure is inverted...
Core dependencies should be in the root and modules are in subdirs.
scripts/core.js
scripts/modules/module1.js
and your problem is solved.
Any further dependency issues will be indicative of defective 'class'/dependency design.

Similar to Mendy, but I create combined files on server-side. The created files will also be minified, and will have a unique name to omit cache issues after an update.
Of course, this practice only makes sense in a whole application or in a framework.

I think your best bet if at all possible, would be to redesign to not have a huge number of javascript files with interfile dependencies. Javascript just wasn't intended to go there.

This is probably too obvious but have you looked at the mootools Core Depender: http://mootools.net/docs/more/Core/Depender

One way to break the parse-time or load-time dependencies is with Self-Defining Objects (a variation on Self-Defining Functions).
Let's say you have something like this:
var obj = new Obj();
Where this line is in someFile.js and Obj is defined in Obj.js. In order for this to parse successfully you must load or concatenate Obj.js before someFile.js.
But if you define obj like this:
var obj = {
init: function() {
obj = new Obj();
}
};
Then at parse or load time it doesn't matter what order you load the two files in as long as Obj is visible at run-time. You will have to call obj.init() in order to get your object into the state you want it, but that's a small price to pay for breaking the dependency.
Just to make it clearer how this works here is some code you can cut and paste into a browser console:
var Obj = function() {
this.func1 = function ( ) {
console.log("func1 in constructor function");
};
this.init = function () {
console.log("init in constructor function");
}
};
var obj = {
init: function() {
console.log("init in original object");
obj = new Obj();
obj.init();
}
};
obj.init();
obj.func1();
And you could also try a module loader like RequireJS.