Replace div with same div onclick with jQuery

First, always make sure your primary code is INSIDE the DOM ready structure, like this:

jQuery(document).ready(function() {
    // This fires when the DOM has loaded (thus all your link(stylesheet) and script includes at the very least
});

Inside your function, I see you using ('#dynamic'), wheras it is missing the $ jQuery global or the jQuery namespace itself, use it like so:

jQuery('#dynamic');

Or

$('#dynamic');

Next, you've got a button you'd want to use to replace content inside a div?

$('button').click(function() {
    $('#dynamic').load('single.php#partial_index');
});

You can also REPLACE your content with a new element, like so:

$('#dynamic').replaceWith($('<div>I am a new jQuery dynamic element!</div>'));

What this code does now is bind a click handler to the document. When the document is clicked, it loads single.php #partial_index into the #dynamic element (though it's missing the $: $('#dynamic').

Here's what I think you want:

$('#dynamic').click(function() {
    $('#test').load("single.php #partial_index");
});

Untested but something like this should work

$("#dynamic").click(function() {
    $("<div />").attr("id", "#partial_index").load("single.php #partial_index", function(data){
         var newDiv = $(this).html(data);
         $('#test').replaceWith(newDiv)  
     })
});