Mini lecture on conservative vector fields and the Fundamental Theorem of Line Integrals

We explain conservative vector fields, using the explanation where f(x,y) is the elevation at the point (x,y) and thus the gradient vector gives the direction of greatest ascent. We explain the Fundamental Theorem of Line Integrals, that it says that the vector line integral of the gradient gives you the total amount you go up, which is your net elevation change f(b) - f(a). This is another form of the Fundamental Theorem of Calculus, and we explain the connection.

[waverly]

I've been learning how to fly on XPlane11 flight simulator. Unlike a car, planes fly along direction vectors between waypoints. The Navigation screen shows you what vector path you are on. (watch the you-tube video "FlyNav--Navigation Display tutorial". That's pretty cool and it helps me to think about vectors.