this command:
db.collection.update(

{ <query selector> }

, { <update operator>:

{ "array.$.field" : value }

} )
use array.$.field" as BasicDBObject key, the exception say:

Exception in thread "main" java.lang.IllegalArgumentException: Document field names can't have a . in them. (Bad Key: 'dimensionArray.$.v')
at com.mongodb.DBCollection.validateKey(DBCollection.java:1843)
at com.mongodb.DBCollection._checkKeys(DBCollection.java:1803)
at com.mongodb.DBCollection._checkObject(DBCollection.java:1790)
at com.mongodb.DBCollectionImpl.update(DBCollectionImpl.java:256)
at com.mongodb.DBCollection.update(DBCollection.java:191)
at com.mongodb.DBCollection.update(DBCollection.java:224)