Should I email a potential supervisor who agreed to support me? I am applying for a PhD at a certain university that requires the student to provide and evidence of support from a potential supervisor before submitting the application. After mailing some professors, one of them agreed to support my application. It was a very brief email saying "I am happy to support you". I immediately prepared a research proposal and sent it to her. She replied by saying "Noted". I prepared my application and submitted it two months later.

Is it appropriate to email her as a reminder of myself? If so, what would be a good email? This university is very important to me and I want to do everything I can to increase my chances of admission.
Yes, in general it's entirely appropriate to remind faculty members about letters/recommendations/paperwork they've previously agreed to.  

In fact many faculty members I've known (and I cannot, alas, wholeheartedly exclude myself) rely on such reminders to a greater or lesser extent.  Faculty nowadays are faced with so much paperwork / routine correspondence (and are not necessarily trained or culturally aligned to regard such things as being important) that we start to believe also in the converse: what we have heard about once and never been reminded of must not be very important.  

Just now I got a second email from a secretary in my department about an annual inventory issue.  She sent the first email just a few days before and included a hard deadline in both emails. In other words, she knows all the tricks to get faculty to do things without causing resentment.  The keys: be polite but persistent, and make the convergence to the deadline clear by ramping up the email reminders accordingly.  It really works: pardon me while I run down to her office so she can tag my laptop!