No. That's the way things work. Also if what you call "client script" is "client.php", then it's not a client side script at all - it's a server side script.
This is how it works
1. Server side script is run. It may produce output, but does not have to. In the particular case of creating web pages however, it will do just that.
2. Output (in the form of HTML code, javascript etc) is sent to the browser.
3. Server side script terminates. It does not remain resident in memory, it has no idea of whom it sent what to. It no longer exists. That is, unless there are persistent side effects (such as writing a file to disk, saving something in a database etc) there will be no remaining evidence that the script ever ran.
The output sent from the server is input for the browser, which parses the code sent and creates the web page. If the code contains javascript code, this is called client side script. The difference is not wether the script does something with the server, or if it sends something to the browser. Code run on the server is server side, code run in the browser is client side.
The browser may send a NEW request to the server, for example by posting a form. But do note that this is indeed a NEW request which has nothing to do with the old one. At least not to the server which has no idea that this is a followup from the previous request (which the server doesn't remember or keep track of).
Thus, you will have to ask yourself what it is you need to retain between the first and second requests, and store that data in some way. Or you could perhaps forgo the second request to the server and do everything you need in the browser using javascript.
To keep data between requests, you have 3 options.
1. Sessions.
Before any output is sent (including a simple whitespace or linebreak before the first <?php opening tag!) use [man]session_start[/man]. After that, anything you assign to the $SESSION array on the server is specific to this user. If you call session_start() on subsequent page requests, the server will know which user it is and restore the $SESSION data. The way this is done is that the server assigns a unique value to the client in the form of a session cookie (which is stored in the browser and sent on every subsequent request).
Query string parameters
The url consists of different parts: protocol://domain.com/path/to/file.php?query=string#hash
The query string comes after the question mark and consists of name/value pairs. Each name/value pair is separated from the previous by & (which has to be sent from the server to the browser as & since & is a special character for the browser. These name/value pairs end up in the $_GET array with the name as key and value as value.
Clientside storage
There are different ways of storing things in the browser, with the most commong probably being cookies. These are domain specific and are always sent on every single request to that domain, although an optional path can be specified as well to restrict when they are sent. They end up in the $_COOKIE array.
But like I said before, you may not even need to send anything to the server after the first request. If all you want to achieve is adding two numbers, why do that on the server? Well, there may very well be reasons for doing so, such as making sure the user isn't changing things. Otherwise you could let the server get the initial request and send the HTML code for the form input controls, as well as send javascript to handle the calculations and presentation of results directly in the browser.
