40A Cooktop on 30A breakers
Let's start with the easy answer. If your installation manual is calling for a 40A circuit for your cooktop and you should install it per their directions. This is because the manufacturers engineers have determined that the recommended circuit is necessary for the cooktop to operate properly. Failure to do so could void the manufacturer's warranty if there is a problem with the cooktop. 

Now the more confusing part. NEC Article 220.55 covers calculations for residential cooking appliances and it does allow derating of these circuits. For example if you have a range rated at 12K you can install a circuit rated for 8k. You can run a single circuit for more than one cooking appliance and tap the circuit for the reduced load of each appliance. I have to say that it is one of the most confusing and complex calculations mentioned in the NEC. 

How can you be sure that what you have will work with your new cooktop? 

You are on the right track by reading your manual and trying to determine the correct KVA to your cooktop, but what you really need to find is the Nameplate rating of this unit. It usually is not in your Installation Manual and it may be hard to find on the unit itself but it will be there somewhere. Another place to find it is to go online to the manufacturer and see if there is a specification sheet for the unit. On the nameplate you will find it's operating voltage and amperage or the required KVA. Then you will have to take that rating and calculate the demand from NEC 220.55. Personally I think that it needs to be supervised by a professional Master Electrician or Engineer or a very competent Journeyman.

Hope this helps, good luck and stay safe.