Planning Energy-Efficient Bike Delivery Routes Through Your City

You recently started a part-time job delivering food by bicycle in your city. The city has varying terrain with hills and valleys, and you want to understand how much energy you expend on different routes. Your friend, who is studying physics, mentions that gravitational force is conservative, meaning the work done by gravity depends only on your starting and ending elevations—not the specific streets you take.

To optimize your delivery routes, you model the net force field acting on you and your bike as [latex]\mathbf{F}(x, y) = \langle 6xy^2, 6x^2y \rangle[/latex], where [latex]x[/latex] and [latex]y[/latex] are measured in kilometers from the city center. You need to verify whether this force field is conservative, find its potential function, and calculate the work done along specific delivery routes. Understanding these concepts will help you choose routes that conserve your energy for a long shift.