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How to get the k-th element in a jQuery set ... AS A JQUERY ELEMENT

So I encounter this situation often and I want to figure out a better way of going about it. I using the naming convention prefixing my jQuery elements with $ so that I can distinguish them from DOM elements. For example, the following:
$pointers = $('#pointer-box div.pointer');
The problem with that is, if I want to get the 3rd in the result as a jQuery, I have to do
$($pointers[2])
which looks awkward. Unless there's a better way of doing this?

You can use .eq() jQuery method
$pointers = $('#pointer-box div.pointer');
var element=$pointers.eq(2);
Given a jQuery object that represents a set of DOM elements, the .eq()
method constructs a new jQuery object from one element within that
set. The supplied index identifies the position of this element in the
set.

Do the property names in JQuery have different names?

I once saw a text about Jquery stating that in Jquery some properties have different names. I think it was the value property that is accessed in Jquery as val or something like that. Does JQuery do this a lot? Is a common practice of Jquery to change properties names?

jQuery does not change any property names.
jQuery implements it's own methods on jQuery objects. Those are completely separate from DOM properties or DOM methods.
For example, you can retrieve the value of any input control from a jQuery object using the .val() method. Internally, the .val() method on the jQuery object accesses the contents of the DOM object (perhaps using the .value property), but it is not replacing that DOM property in any way. It is still there is you choose to use it instead.
Similarly, jQuery has a .html() method that returns the innerHTML of an object. Again, this is jQuery method on a jQuery object and does not replace the DOM property innerHTML in any way.
Here are some code examples to illustrate:
<input id="test" type="text">
<div id="title">This is my Title</div>
Using plain javascript:
var text = document.getElementById("test").value;
var title = document.getElementById("title").innerHTML;
Using jQuery:
var text = $("#test").val();
var title = $("#title").html();
Notice that both the jQuery ways are method calls, not properties.
Even when using jQuery objects, you could get the DOM object out of the jQuery object using the .get(n) method and use the DOM property, though there usually isn't a reason to do so:
var text = $("#test").get(0).value;
var title = $("#title").get(0).innerHTML;

jQuery doesn't change property names. jQuery returns jQuery instances not dom elements.
If you do $thing[0].value it will work. If you do $thing.prop("value") it will work.
If you use $thing.val(); which is jQuery's val method (which is "kind" of like val but does a bunch of weird edge case magic as well) then it will "probably" return the value of the wrapped dom element.

Not that I'm aware of, the case of val is because it let's you set or get the value of all form fields (textfields, radio buttons, checkboxes, textareas, etc.) something that's not possibly simply by doing input.value in all cases.

document.getElementById vs jQuery $()

Is this:
var contents = document.getElementById('contents');
The same as this:
var contents = $('#contents');
Given that jQuery is loaded?

Not exactly!!
document.getElementById('contents'); //returns a HTML DOM Object
var contents = $('#contents'); //returns a jQuery Object
In jQuery, to get the same result as document.getElementById, you can access the jQuery Object and get the first element in the object (Remember JavaScript objects act similar to associative arrays).
var contents = $('#contents')[0]; //returns a HTML DOM Object

No.
Calling document.getElementById('id') will return a raw DOM object.
Calling $('#id') will return a jQuery object that wraps the DOM object and provides jQuery methods.
Thus, you can only call jQuery methods like css() or animate() on the $() call.
You can also write $(document.getElementById('id')), which will return a jQuery object and is equivalent to $('#id').
You can get the underlying DOM object from a jQuery object by writing $('#id')[0].

Close, but not the same. They're getting the same element, but the jQuery version is wrapped in a jQuery object.
The equivalent would be this
var contents = $('#contents').get(0);
or this
var contents = $('#contents')[0];
These will pull the element out of the jQuery object.

A note on the difference in speed. Attach the following snipet to an onclick call:
function myfunc()
{
var timer = new Date();
for(var i = 0; i < 10000; i++)
{
//document.getElementById('myID');
$('#myID')[0];
}
console.log('timer: ' + (new Date() - timer));
}
Alternate commenting one out and then comment the other out. In my tests,
document.getElementbyId averaged about 35ms (fluctuating from 25ms up to 52ms on about 15 runs)
On the other hand, the
jQuery averaged about 200ms (ranging from 181ms to 222ms on about 15 runs).
From this simple test you can see that the jQuery took about 6 times as long.
Of course, that is over 10000 iterations so in a simpler situation I would probably use the jQuery for ease of use and all of the other cool things like .animate and .fadeTo. But yes, technically getElementById is quite a bit faster.

No. The first returns a DOM element, or null, whereas the second always returns a jQuery object. The jQuery object will be empty if no element with the id of contents was matched.
The DOM element returned by document.getElementById('contents') allows you to do things such as change the .innerHTML (or .value) etc, however you'll need to use jQuery methods on the jQuery Object.
var contents = $('#contents').get(0);
Is more equivilent, however if no element with the id of contents is matched, document.getElementById('contents') will return null, but $('#contents').get(0) will return undefined.
One benefit on using the jQuery object is that you won't get any errors if no elements were returned, as an object is always returned. However you will get errors if you try to perform operations on the null returned by document.getElementById

No, actually the same result would be:
$('#contents')[0]
jQuery does not know how many results would be returned from the query. What you get back is a special jQuery object which is a collection of all the controls that matched the query.
Part of what makes jQuery so convenient is that MOST methods called on this object that look like they are meant for one control, are actually in a loop called on all the members int he collection
When you use the [0] syntax you take the first element from the inner collection. At this point you get a DOM object

In case someone else hits this... Here's another difference:
If the id contains characters that are not supported by the HTML standard (see SO question here) then jQuery may not find it even if getElementById does.
This happened to me with an id containing "/" characters (ex: id="a/b/c"), using Chrome:
var contents = document.getElementById('a/b/c');
was able to find my element but:
var contents = $('#a/b/c');
did not.
Btw, the simple fix was to move that id to the name field. JQuery had no trouble finding the element using:
var contents = $('.myclass[name='a/b/c']);

var contents = document.getElementById('contents');
var contents = $('#contents');
The code snippets are not the same. first one returns a Element object (source).
The second one, jQuery equivalent will return a jQuery object containing a collection of either zero or one DOM element. (jQuery documentation). Internally jQuery uses document.getElementById() for efficiency.
In both the cases if more than one element found only the first element will be returned.
When checking the github project for jQuery I found following line snippets which seems to be using document.getElementById codes (https://github.com/jquery/jquery/blob/master/src/core/init.js line 68 onwards)
// HANDLE: $(#id)
} else {
elem = document.getElementById( match[2] );

Just like most people have said, the main difference is the fact that it is wrapped in a jQuery object with the jQuery call vs the raw DOM object using straight JavaScript. The jQuery object will be able to do other jQuery functions with it of course but, if you just need to do simple DOM manipulation like basic styling or basic event handling, the straight JavaScript method is always a tad bit faster than jQuery since you don't have to load in an external library of code built on JavaScript. It saves an extra step.

One other difference: getElementById returns the first match, while $('#...') returns a collection of matches - yes, the same ID can be repeated in an HTML doc.
Further, getElementId is called from the document, while $('#...') can be called from a selector. So, in the code below, document.getElementById('content') will return the entire body but $('form #content')[0] will return inside of the form.
<body id="content">
<h1>Header!</h1>
<form>
<div id="content"> My Form </div>
</form>
</body>
It might seem odd to use duplicate IDs, but if you are using something like Wordpress, a template or plugin might use the same id as you use in the content. The selectivity of jQuery could help you out there.

All the answers are old today as of 2019 you can directly access id keyed filds in javascript simply try it
<p id="mytext"></p>
<script>mytext.innerText = 'Yes that works!'</script>
Online Demo!
- https://codepen.io/frank-dspeed/pen/mdywbre

jQuery is built over JavaScript. This means that it's just javascript anyway.
document.getElementById()
The document.getElementById() method returns the element that has the ID attribute with the specified value and Returns null if no elements with the specified ID exists.An ID should be unique within a page.
Jquery $()
Calling jQuery() or $() with an id selector as its argument will return a jQuery object containing a collection of either zero or one DOM element.Each id value must be used only once within a document. If more than one element has been assigned the same ID, queries that use that ID will only select the first matched element in the DOM.

All the answers above are correct. In case you want to see it in action, don't forget you have Console in a browser where you can see the actual result crystal clear :
I have an HTML :
<div id="contents"></div>
Go to console (cntrl+shift+c) and use these commands to see your result clearly
document.getElementById('contents')
>>> div#contents
$('#contents')
>>> [div#contents,
context: document,
selector: "#contents",
jquery: "1.10.1",
constructor: function,
init: function …]
As we can see, in the first case we got the tag itself (that is, strictly speaking, an HTMLDivElement object). In the latter we actually don’t have a plain object, but an array of objects. And as mentioned by other answers above, you can use the following command:
$('#contents')[0]
>>> div#contents

How does $(this) work in jQuery

How does the jQuery tag $(this) exactly work? I know how to use it, but how does jQuery know which element is 'active'? And what is the original Javascript tag for getting the current item, or is it jQuery only?

The this is a simple javascript (DOM) object,
$(this) will turn the object into a jQuery object.
jQuery doesn't need to 'know' what this is, it doesn't treat this in a special way, no other than myHeaderDiv in
var myHeaderDiv = document.getElementById('header');
$myHeaderDiv = $(myheaderDiv); //just a variable transformed into jQuery object, as with this.

this is context-dependent in jQuery (and JavaScript in general). It usually represents the current DOM element in a event handler, but is not a jQuery object.
$(this) is a jQuery object containing the current DOM element.

The expression $(this) is just a regular Javascript function call, equivalent to jQuery(this). The this value is defined by Javascript itself, and is not a jQuery invention.

Perhaps you should read about scope in JavaScript
http://www.digital-web.com/articles/scope_in_javascript/

Jquery - Reference by ID - Supposed to return an array?

I just started using jQuery, and various sources suggest that the following should be used to reference an element by ID:
$("#imgThumbnail")
theoretically making something like this possible:
$("#imgThumbnail").src;
But my testing indicates that something like $("#imgThumbnail") returns an array, making the following necessary:
$("#imgThumbnail")[0].src;
Do I really need to reference by the index of the array every time I am trying to reference something by ID (i.e., var oObj = $("#someobjectid")[0]; )?

You should get the src attribute to get the value
$("#imgThumbnail").attr('src');

This post explains what the $ function returns and various ways to use it.
$(selector)
Returns a jQuery object, which could contain a number of DOM elements.
$(selector)[0] or $(selector).get(0)
Returns the first result as an actual DOM element.
$(selector).eq(0) or $($(selector).get(0))
Returns the DOM element wrapped in a jQuery object so that we can do stuff like:
$(selector).eq(0).addClass("deleted").fadeOut();

$(specifier) will return a collection, so yes if you want to call something on an individual member you need to pick which one. In most cases though there is a collection operator you can use to achieve the same result. For instance, you could call $('#imgThumbnail').attr('src', 'value')

You should bear in mind that it's not really an array, it's a jQuery object which, among other things, allows array-style access

$(whatever)
returns the jQuery object. On the jQuery object you can do jQuery and jQuery plugin things, eg. .text() to return the text inside the element or .css("background", "pink") to make the element(s) pink.
Since src isn't a jQuery thing you cannot access it. src is however both a HTML attribute, and you can access those with the attr method:
.attr("src")` and `.attr("src", "http://www.example.com/myimage.png")
src is also a DOM-property and you can access DOM-properties using [index] or by iterating through the jQuery object with each:
.each(function(){
this.src = "http://www.example.com/myimage.png";
})

I don't think you should be using .src with jQuery.
Try $("#imgThumbnail").attr('src');
(this will read the src attribute, you set it with a second arg if you like)
See here:
http://docs.jquery.com/Attributes/attr

to set the src attribute use
$("#imgThumbnail").attr("src", value)
if you use something like a class selector or tag like so
$("img").attr("src", value)
It will modify all the image src attributes on the page. Hence the $ function returns an array.
And you do not need to reference it specifically.