Please share HTML of your view. First of All you need 3 different dropdown that hopefullly will be there as you did not post your HTML.
PartialViews has to reloaded with javascript and to achieve this partialview should be in some container like
<div id="partialview">@{Html.RenderPartial("yourPartialView",modelobject,viewdatadictionary)}</div>
then in javascript you have to reload that container div of your partialview.
$("#partialview").load("@Html.Action("ActionName","ControllerName",querystring values)");
in your case:
$("#CategoryId").change(function () {
var id = $(this).val();
$("#partialview").load("@Html.Action("CategoryList","Manage",new {catId=id})");
your partial view will be reloaded acording to your need.