According to  U.S. (CBP), if a child is traveling abroad with one parent, the other  parent must supply a notarized letter granting permission. If the child  is traveling with an adult other than her parents, both parents must  provide written permission.
While U.S. customs officials do not  always ask for this documentation, many countries do, and failure to  produce notarized permission letters could result in being denied entry.
Requirements
Many  parents aren’t familiar with notarial rules and procedures, and may ask  to notarize an absent spouse’s signature, not realizing it’s against  the law. We cannot do That, Both parents must personally appear before  the notary.
Notaries Can't Give Advice.
As a Notary, we cannot  give legal advice regarding the contents of a permission letter or how a  document must be completed. If the signer has questions, they will need  to contact the company or agency requiring the document for further instructions.
Plan ahead.
International travelers often rush and  panic at the last minute to make sure the necessary paperwork is in  order prior to imminent departure. Parents who need notarized permission  for a child to travel at the last minute may get flustered if their signatures cannot be notarized due to lack of acceptable ID or another  issue.  
Inspired from David Thun an Associate Editor at the National Notary Association