What are clear examples that connect Goethe's Faust to Rousseau's romantic ideals?

Since Rousseau didn't publish his first work,
New Heloise, until 1760 and Goethe finished what is called the
Urfaust in 1775, following The Sorrows of Young Werther,
it must be stated that any connection between the two would not correctly
reflect influence. In other words, Rousseau cannot be assumed to have influenced
Goethe's work even though Goethe only completed and published Faust Part
I in 1808, and this he only did at friends urgent promptings. The reason is
that, having begun Romanticism with Werther, Goethe soon renounced
Romanticism, after having written the Urfaust, as a result of
seeing the end of a young woman's suicide; as she was pulled out of a frozen river,
Goethe saw a copy of Werther in her pocket. Thus he abandoned the
Romanticist Faust until urged to complete it, which he did in a
neo-classical style, not a Romantic style. Yet it may be possible to say that the two
connect because Goethe countered the tenets of Romanticism in Faust Part
I.

Having said this, some commonalities between
Faust and Rousseau's ideas relate to (1) emotionalism and (2)
nature.

1. At first, Faust
tells Mephistopheles that he can't be tempted by human emotions. Then he switches to the
position that he has exhausted academic study therefore will immerse himself in the
passions that make people so earnestly moved and motivated. As a result of this switch,
Faust demands Mephisto procure Margarete for him. 

One of
Rousseaus's prominent ideas is that people are good in
their innate emotions and passions and that it is civilizing forces of society and
uniformizing forces in education that corrupt these emotions and passions. Thus there is
a direct connection between Rousseau's ideas and the sensations Faust pursues in Part I.
In addition, this connection is dramatized by Margarete/Gretchen's ravings in the
dungeon and her exchange with Faust there. The connection may not be viewed as a
supportive one, however. Goethe is calling this sort of emotionalism and passion into
question, especially as pursued by Faust (less so that natural emotion and passion
demonstrated by Gretchen).

2. Another connection is in
"Gloomy Day" in which Faust is in a nature setting after
his seduction of Gretchen. Mephisto has orchestrated events so that Faust slays
Gretchen's brother Valentine. As a result, Faust has to flee to escape arrest. Mephisto
deliberately acted to leave Gretchen abandoned, pregnant and unmarried. Thus Faust is
living contentedly in nature, knowing nothing of the harm and tragedy that is unfolding
in the city--from Gretchen's pregnancy to her mother's death to, eventually, her baby's
death. Here Goethe seems to be suggesting that while nature is restorative and peaceful,
it is an escape from the realities of human life, which of necessity includes
interactions and community.

One of
Rousseau's ideas is that nature is good and simple while
city life is corrupt and artificial. He believed that people should leave the towns and
leave behind the laws and institutions and return to a state of living peacefully in
nature. His idea that nature is kindly and good had a powerful effect on European and
English Romanticism as is evidenced by its emphasis on the pastoral lifestyle. This idea
connects to Faust because Faust is sent by Mephisto to nature to
escape the institutions and laws of towns and to regenerate from his ordeal with
Valentine. Yet the connection fails when Faust learns of Gretchen's
plight.