How to connect to LocalDb

I installed LocalDb using the SqlLocalDb.msi package and I can connect to it using SSMS using the server name (LocalDb)\v11.0. So far so good. The problem is that when I try to connect to it via a .NET 4.5 application I get the error

The server was not found or was not accessible.

I started with the connection string Data Source=(LocalDb)\v11.0 and then added many things to no avail (integrated security true/sspi, attaching a db, using instances, etc.)

I am totally unable to connect to localdb with any tool including MSSMA, sqlcmd, etc. You would think Microsoft would document this, but I find nothing on MSDN. I have v12 and tried (localdb)\v12.0 and that didn't work. Issuing the command sqllocaldb i MSSQLLocalDB shows that the local instance is running, but there is no way to connect to it.

c:\> sqllocaldb i MSSQLLocalDB
Name:               MSSQLLocalDB
Version:            12.0.2000.8
Shared name:
Owner:              CWOLF-PC\cwolf
Auto-create:        Yes
State:              Running
Last start time:    6/12/2014 8:34:11 AM
Instance pipe name: np:\\.\pipe\LOCALDB#C86052DD\tsql\query
c:\>
c:\> sqlcmd -L

Servers:
    ;UID:Login ID=?;PWD:Password=?;Trusted_Connection:Use Integrated Security=?;
*APP:AppName=?;*WSID:WorkStation ID=?;

I finally figured it out!! the connect string is (localdb)\MSSQLLocalDB, e.g.:

$ sqlcmd -S \(localdb\)\\MSSQLLocalDB
1> select 'hello!'
2> go

------
hello!

(1 rows affected)    

Use (localdb)\MSSQLLocalDB. That is the LocalDB instance intended for applications, independent of Visual Studio version.

Disregard my original answer: "With SQL Server 2014 Express LocalDB, use (localdb)\ProjectsV12. This works in both Visual Studio 2013 and SQL Server 2014 Management Studio." While ProjectsV12 will indeed give you a LocalDB instance, it's the wrong one, intended for use by SQL Server Data Tools.