How do I call Java methods from handwritten JavaScript or third party libraries?

You can define a "bridge" method via JSNI that provides an external, globally visible JavaScript function that can be called by your hand-crafted JavaScript code. For example,

package mypackage;
public MyUtilityClass
{
    public static int computeLoanInterest(int amt, float interestRate, int term) { ... }
    public static native void defineBridgeMethod() /*-{
       $wnd.computeLoanInterest = function(amt, intRate, term) {
          return @mypackage.MyUtilityClass::computeLoanInterest(IFI)(amt, intRate, term);
       }
    }-*/;
}

On application initialization, call MyUtilityClass.defineBridgeMethod() (e.g. from your GWT Entry Point). This will create a function on the window object called "computeLoanInterest" which will invoke, via JSNI, the compiled Java method of the same name. The bridge method is needed because the GWT compiler will obfuscate/compress/rename the names of Java methods when it translates them to JavaScript.