Multiple submit button validation in jquery

I have added 4 buttons in my Html Form. and I have to add different validation and action for that button. when I attend to validate button That didn't work.,and also I have to add different action for the other buttons.May any one please Help Me..

<form id="form-sign-up" class="styled" action="DepartmentServlet" method="post">
            <fieldset>
                <h3>Department </h3>
                <ol>
                    <li class="form-row"><label>Department Id:</label>
                        <input type="text" name="departmentId" id="departmentId" class="text-input required"/><span id="errmsg"></span>
                    </li>
                    <li class="form-row"><label>Department Name:</label>
                        <input type="text" name="departmentName" id="departmentName" class="text-input required"/>
                    </li>
                    <li class="form-row"><label>Department Head:</label>
                        <select name="departmentHead" class="text-input required" id="myDropdown"style="width: 158px"> <option value="">Select Department Head</option>
                            <%

                                String val = "";
                                Department dp = new Department();
                                ResultSet rs = dp.getHeadOfDepartment();
                                while (rs.next()) {
                                    val = rs.getString(2);
                            %>

                            <option value="<%= val%>"><%= val%></option>
                            <%
                                }

                            %>


                        </select>
                    </li>
                    <li class="form-row"><label>Appointed Date:</label>
                        <input id="demo1" name="appointedDate"type="text" maxlength="25" size="25" class="text-input required"><a href="javascript:NewCal('demo1','ddmmmyyyy',true,24)"><img src="images/cal.gif" width="16" height="16" border="0" alt="Pick a date"></a>
            <span class="descriptions">pick a date..</span>
                    </li>

                    <li class="button-row">
                       <input type="submit" value="ADD" class="btn-submit img-swap"/>
                       <input type="submit" value="Delete" class="btn-delete img-swap"/>
                       <input type="submit" value="Update" class="btn-update img-swap"/>
                       <input type="submit" value="Search" class="btn-search img-swap"/>
                    </li>
                </ol>
            </fieldset>

$(document).ready(function() { 
     $("#departmentId").keypress(function (e)  
    { 
      //if the letter is not digit then display error and don't type anything
      if( e.which!=8 && e.which!=0 && (e.which<48 || e.which>57))
      {
        //display error message
        $("#errmsg").html("Digits Only").show().fadeOut("slow"); 
        return false;
      } 
    });
      $('.btn-delete').click(function(e){
          var $error = $('<span class="error"></span>');

         if (("#departmentId".val() == "")) {

             $parentTag.addClass('error').append($error.clone().text('Required Field'));

         }


     });

    $('.btn-submit').click(function(e){

        // Declare the function variables:
        // Parent form, form URL, email regex and the error HTML
        var $formId = $(this).parents('form');
        var formAction = $formId.attr('action');
        var emailReg = /^([\w-\.]+@([\w-]+\.)+[\w-]{2,4})?$/;
        var $error = $('<span class="error"></span>');

        // Prepare the form for validation - remove previous errors
        $('li',$formId).removeClass('error');

        $('span.error').remove();

        // Validate all inputs with the class "required"
        $('.required',$formId).each(function(){
            var inputVal = $(this).val();
            var $parentTag = $(this).parent();
            if(inputVal == ''){
                $parentTag.addClass('error').append($error.clone().text('Required Field'));
            }






            // Run the email validation using the regex for those input items also having class "email"
            if($(this).hasClass('email') == true){
                if(!emailReg.test(inputVal)){
                    $parentTag.addClass('error').append($error.clone().text('Enter valid email'));
                }
            }

            // Check passwords match for inputs with class "password"
            if($(this).hasClass('password') == true){
                var password1 = $('#password-1').val();
                var password2 = $('#password-2').val();
                if(password1 != password2){
                    $parentTag.addClass('error').append($error.clone().text('Passwords must match'));
                }
            }

        });

        // All validation complete - Check if any errors exist
        // If has errors



        if ($('span.error').length > 0) {

            $('span.error').each(function(){

                // Set the distance for the error animation
                var distance = 5;

                // Get the error dimensions
                var width = $(this).outerWidth();

                // Calculate starting position
                var start = width + distance;

                // Set the initial CSS
                $(this).show().css({
                    display: 'block',
                    opacity: 0,
                    right: -start+'px'
                })
                // Animate the error message
                .animate({
                    right: -width+'px',
                    opacity: 1
                }, 'slow');

            });
        } else {
            $formId.submit();
        }
        // Prevent form submission
        e.preventDefault();


    });

    // Fade out error message when input field gains focus
    $('.required').focus(function(){
        var $parent = $(this).parent();
        $parent.removeClass('error');
        $('span.error',$parent).fadeOut();
    });



});