Small fixes to contributing documentation.

 - Using widgets/spinning_square.dart instead of non-existing
  examples/widgets for the `run -t` example
 - travis/test.sh -> dev/bots/test.sh
 - Couple of style changes.

For developers with the platform tools installed but no SDK, it prevents
crashing in the flutter tool in favor of more user-friendly error messages
downstream.

* New Material Design calculator example. (Attempt 2 after git fetch upstream).

* Responded to code review.

* Second round of code review.

Rev the dart sdk to 1.17.0-dev.4.1

Keeps up with the Joneses, pulling in the latest dev build (and with it a number of stability and perf improvements).

Looks like our intent here was to show the tab label rather than
"Instance of '_Page'".

Also, fix an issue where the scrollableKey wasn't being forwarded to the
actual scrollable.

Fixes #3090

Fix analyzer field overrides.

Cleans up 3 of the 4 violations of the `overriden_field` lint.

The last one is more interesting and I'll defer to someone closer to that code:

    [lint] Do not override fields. (packages/flutter/test/rendering/rendering_tester.dart, line 91, col 14)

@chinmaygarde

* rebuild when the build type changes

* handle local engine builds

* also check the timestamp

Cleanup dangling comment references.

Quick pass at fixing a few dangling references as revealed by the new `comment_references` lint (https://github.com/dart-lang/linter/issues/240).

There's a bunch more to do here before we can turn it on by default (~430 lints as of now).  Many of them are a simple matter of adding an import (e.g., `dart:async` for library docs that reference `Future`) but others will require a bit of thought.  Probably best done by the folks writing the code. :)

Don't suggest filing an issue when we can definitively say that the
assertion in question was not thrown from within the flutter package.
Fixes https://github.com/flutter/flutter/issues/3812.

Put the stack trace first after the message, with more details below the
stack trace, since the stack is often very useful and the stacks are no
longer stupidly long.

Better document the 'rescheduling' feature, and improve the error
handling asserts when it is misused.
Fixes https://github.com/flutter/flutter/issues/3888.

Improve the wording around the stack dump for exceptions during
callbacks.

Improve the color and font size of messages in live tests.
Fixes https://github.com/flutter/flutter/issues/4018.

This patch updates the date picker to match the new spec.

We're still missing a fade effect when scrolling from one month to
another and we're missing the landscape layout.

Fixes #3558

Update to new analyzer and linter.

Notably, this will allow us to play with

* the fixed `public_member_api_docs` that now checks for documented getters when checking setters (https://github.com/dart-lang/linter/issues/237), and
* the new `comment_references` lint that ensures identifiers referenced in docs are in scope (https://github.com/dart-lang/linter/issues/240).

This reverts commit f91cf1c4.

Notably, this will allow us to play with

* the fixed `public_member_api_docs` that now checks for documented getters when checking setters (https://github.com/dart-lang/linter/issues/237), and
* the new `comment_references` lint that ensures identifiers referenced in docs are in scope (https://github.com/dart-lang/linter/issues/240).

This moves all the bot-related files to `dev/bots`, hiding it from our
home page in github. Also, simplifies the travis setup, though that
doesn't do any difference to the performance sadly.

The latest engine roll caused a performance regression: https://github.com/flutter/flutter/issues/4005

Also, clean up the DatePicker to use more modern technology, such as a
grid for displaying the days of the month.

Fixes #3976

Also, make some previously private classes public for better
documentation.