Conservative vector fields
- Fundamental theorem of line integrals, also known as the gradient theorem.
What we're building to
- A vector field is called a conservative vector field if it satisfies any one of the following three properties (all of which are defined within the article):
- Line integrals of are path independent.
- Line integrals of over closed loops are always .
- is the gradient of some scalar-valued function, i.e. for some function .
- There is also another property equivalent to all these: is irrotational, meaning its curl is zero everywhere (with a slight caveat). However, I'll discuss that in a separate article which defines curl in terms of line integrals.
- The key takeaway here is not just the definition of a conservative vector field, but the surprising fact that the seemingly different conditions listed above are equivalent to each other. Madness!
Definition: This property is called path independence. Specifically, a line integral through a vector field is said to be path independent if the value of the integral only depends on the point where the path starts and the point where it ends, not the specific choice of path in between.
Path independence implies gradient field
Definition: A path is called closed if it starts and ends at the same point. Such paths are also commonly called closed loops.
Funky notation for closed-loop integrals.
- Imagine walking clockwise on this staircase. With each step gravity would be doing negative work on you. So integrating the work along your full circular loop, the total work gravity does on you would be quite negative. However, that's an integral in a closed loop, so the fact that it's nonzero must mean the force acting on you cannot be conservative.
- Imagine walking from the tower on the right corner to the left corner. If you get there along the clockwise path, gravity does negative work on you. If you get there along the counterclockwise path, gravity does positive work on you. Since both paths start and end at the same point, path independence fails, so the gravity force field cannot be conservative.
- In the real world, gravitational potential corresponds with altitude, because the work done by gravity is proportional to a change in height. What makes the Escher drawing striking is that the idea of altitude doesn't make sense. Many steps "up" with no steps down can lead you back to the same point. This corresponds with the fact that there is no potential function such that give the gravity field.