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When in JS you need to identify element by array order?

Ok so I finally have a code example to show this!
if ($('#Snowsports-row')[0].classList.contains("hidden") == false) {
$('#snowsports-only').removeClass("hidden")
}
The code works ONLY as written above, i.e., if the [0] were moved to the second line and removed from the first line, or if it were present/absent in both lines, it would fail.
I understand the output difference...
$('#Snowsports-row')
=> [<div>...]
$('#Snowsports-row')[0]
=> <div>...
...but I'm not understanding under what circumstances you're OK to get an array of element(s) and in which you need to tease out the exact element.
THANKS FOR ALL ANSWERS! Very clearly helped me to figure out that the problem may have been confusing JS/jQuery methods. Final version:
if ($('#Snowsports-row').hasClass("hidden") == false) {
$('#snowsports-only').removeClass("hidden")
}

The .classList method is not widely supported (not in MSIE 9.0 for example) so it's not portable, although where it exists it's fast.
Since every ID in a document is supposed to be unique, and since calling removeClass for a class that isn't present is harmless, just replace your entire call with:
$('#Snowsports-row').removeClass('hidden')
Or better yet, if that class means what I think it does, use .hide() and let jQuery do its job for you, potentially animation the transition in the process.
Alternatively, if you actually wanted to stick with using DOM and classList, you should use the .remove() method that classList already supports:
document.getElementById('#Snowsports-row').classList.remove('hidden')
although there's a minor disadvantage in that this code will crash if that element isn't found (since .getElementById will return null) whereas jQuery silently ignores calls made on empty selectors.
As for the meta-question - you use [n] if you want to access the single DOM element at position n within the jQuery object, as you've done when you use .classList.
You use .eq(n) to obtain a jQuery object representing that DOM element, e.g. if you want to apply jQuery methods to that (single) element.
If there's only a single element, or you want the jQuery method to apply to every matching element, just call the method directly on the selector, as I've done above.

First off, by using jQuery for what it's good at, you can replace this:
if ($('#Snowsports-row')[0].classList.contains("hidden") == false) {
$('#snowsports-only').removeClass("hidden")
}
with this:
$('#Snowsports-row').removeClass("hidden");
Your first block of code does the following:
With $('#Snowsports-row'), make a jQuery object that contains all DOM elements that match the select '#Snowsports-row'.
Then reach into the jQuery object with [0] and get the first DOM object in that jQuery object.
Then, use a property/method on that DOM element to determine if a class exists on that DOM element with your .classList.contains("hidden") reference.
Then, if you find that class, remove it.
A jQuery object contains inside it an array of DOM elements. If you call a method on the jQuery object itself like:
$('.tableRows').html("hello");
Then, you are asking jQuery to operate on ALL the DOM elements inside the jQuery object. You must use jQuery methods, not DOM methods.
If, on the other hand, you want to use a method such as .classList.contains(), that is only a method on an actual DOM element. That isn't a jQuery method. So, you have to reach inside of the jQuery object to get a specific DOM element out of it. That's what the [0] does. It reaches into the jQuery object and gets the first DOM element from its internal data structure. Once you have that DOM element, you can then use any DOM element methods on that DOM object.
FYI, if you ever want to get just the first DOM element from a jQuery object, but want the result to be a jQuery object, not just a DOM element, instead of [0], you can use .eq(0) like ths:
$('#Snowsports-row').eq(0).removeClass("hidden");
Now, in this specific case, this is never necessary because $('#Snowsports-row') cannot ever contain more than one DOM element because internally jQuery will only return the first matching DOM element when you are searching for a ID value (since there's never supposed to be more than one matching element with the same ID).
Just keep in mind that DOM element and a jQuery object are completely different types of objects with different methods on them. What makes it slightly confusing is that a jQuery object contains an internal list of DOM elements. But, if the object you are operating on is a jQuery object, then you can only call jQuery methods on it. If you reach into the jQuery object and pull out a DOM element, then you can only call DOM methods on it.

First of all, ids must be unique, so if you have more than one #Snowsports-only elements you can experience problems.
In your question, you are mixing jQuery code with pure Javascript code.
This:
if ($('#Snowsports-row')[0].classList.contains("hidden") {
...
}
Means that you get the first instance of #Snowsports-row (remember that is better if there is only one element with this id), but you get the DOM object (pure javascript) with the jQuery selector. You can do the same thing in jQuery like this:
$('#Snowsports-row').hasClass("hidden")
See more:
https://api.jquery.com/hasclass/
https://developer.mozilla.org/es/docs/Web/API/Element/classList

Sure, because you are operating over a list. Now, you're kind of mistaking the jQuery/javascript code. If you would like to use the same line twice you can basically drop jQuery altogether and write something like this:
var el = document.getElementById('Snowsports-row');
if (el.classList.contains('hidden')){
el.classList.remove('hidden');
}

In the first line you're selecting one specific DOM element, whereas in the second line you are selecting ALL elements in the DOM that fit that selector and removing the "hidden" class from all of them. Basically checking whether the element has a class can only be performed over an element (that's why you need to select the index, specifying a given element), but jQuery allows you to remove the class of every element inside a list (hence your second line)

Use jQuery's .eq() function. So:
var el = $('#Snowsports-row').eq(0);
if (el.hasClass("hidden")) {
$(el.removeClass("hidden")
}
There's also no harm in calling removeClass on an element that might not have that class... so:
$('#Snowsports-row').eq(0).removeClass('hidden');

Make get() returning a jQuery object

Is there a way to have get() returning a jQuery object instead of "just" the DOM element?
Example:
$("div").get(0) returns [<div></div>] instad of <div></div>.
I'd like to prevent overwrapping like $($("div).get(0)) because the query will get a little bit longer than the example and I fear the readability gets lost. I'd rather not use variables to save unnecessary DOM elements in the RAM, either.

Use eq() to return the jquery object instead of get()
$("div").eq(0)
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, jQuery doc

You would use eq:
$("div").eq(0)

Use eq() instead , which will return jQuery object
$("div").eq(0)

Also try this also:
$('div:first')

javascript alternative for jquery element addressing

What is Javascript alternative for this:
$('#clientDetailModal #heightValue')
I need it because in my code this works:
document.getElementById('heightValue').checkValidity()
but this doesn't:
$('#clientDetailModal #heightValue').checkValidity()
And I need to select only heightValue within clientDetailModal div.

Try $('#clientDetailModal #heightValue')[0].checkValidity()
The reason you need to do the [0] is, (as per the jquery id selector documentation)
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
Since you'll get a collection with 1 DOM element (assuming you don't have multiple ids), you need to then explicitly "select" that element using the [0].

You could use get to get the DOM element :
$('#clientDetailModal #heightValue').get(0).checkValidity()
Just to be sure, as your question might be a little ambiguous : only one element can have a given ID in HTML. So if your element is either absent or inside #clientDetailModal, then you could as well use
$('#heightValue').get(0).checkValidity()
It would also be faster. But in that case, there would be nothing wrong in using document.getElementById.

Since document.getElementById('heightValue').checkValidity() works, it means your function checkValidity() is attached on native DOM elements. This means, you can do:
$('#clientDetailModal #heightValue')[0].checkValidity()
Plus: If your HTML is valid with no duplicate IDs, you can simply do
$('#heightValue')[0].checkValidity()

Since the OP asked for a JavaScript alternative. On modern browsers,
document.querySelector ('#clientDetailModal #heightValue')
will return the element you are asking for.
The direct equivalent would be
document.querySelectorAll ('#clientDetailModal #heightValue')
which returns an array of elements matching the selector requested, do yrou will need to add the [0] as per the other answers.

I presume this is what you're looking for :
document.getElementById('clientDetailModal').getElementById('heightValue').checkValidity();

jQuery - how to access jQuery object?

I'm struggling to find out when to use #,$ in jquery.
i.e. if I have an object
var elem{
}
How to access this?
Whether $('#elem') or $('elem')?
May be its too silly. But I cant find out a solution by googling it.

It looks to me like how to you've used .appendTo(). You need to pass an ID in your case, I believe; something like:
$("<table>").attr("id","waiterBlock").appendTo("testDiv");
Changed to:
$("<table>").attr("id","waiterBlock").appendTo("#testDiv");
The documentation for jQuery's .appendTo() says one needs to pass:
A selector, element, HTML string, or jQuery object; the matched set of elements will be inserted at the end of the element(s) specified by this parameter.

Cant get jQuery methods to work

Why is it when I select html elements using .get(i) or similar methods, I am unable to use methods on those elements like the .removeClass() or .html().
I would think that the code below is perfectly valid, yet neither lines work. What do I need to do to apply jQuery methods to an element based on its ordinal index in the DOM?
($('li').get(0)).removeClass('yourClass');
$('li')[0].addClass('myClass');
Here is an example of the issue: http://jsfiddle.net/KcNWy/2/

Check the documentation:
The .get() method grants us access to the DOM nodes underlying each jQuery object.
You can DOM object into jQuery object again: $($('li').get(0)). Or, better yet, use eq: $('li').eq(0).
And also a debugging hint. You can use in firefox/chrome/safari console.log(myObject) to see what's actually returned.

The get() method passes back the DOM object
Retrieve the DOM elements matched by the jQuery object.
Try this instead:
$("li:eq(" + 0 + ")").removeClass("yourClass")
Working example: http://jsfiddle.net/KcNWy/6/