# Vector Fields, Phase Space, Phase Flow Start by watching [this video](https://youtu.be/rB83DpBJQsE?t=477) by 3b1b. In it he specifically shows how [Divergence](Divergence.md) and [Curl](Curl.md) can be applied not only to physical spaces, but **phase spaces**.  We can describe how our variables (fox and rabbit) change over time via a set of **differential equations**.  A nice way to visualize what these equations are really saying is to visualize each point on the plane with a vector, representing the two derivatives at that point in time:  For example, when there are lots of foxes but relatively few rabbits the number of foxes may tend to go down due to the constrained food supply and the number of rabbits may also tend to go down because they are being eaten by all of the foxes (potentially at a rate that is faster than what they can reproduce):  > This vector field is not about physical space, but rather it a representation of a dynamic system (foxes vs. rabbits) that has two variables, and how that system evolves over time. This provides a good insight into why we want to study the geometry of higher dimensions! What if our system was tracking more than just 2 or 3 numbers? The flow associated with this field is called the **Phase Flow** of our differential equation:  It is a way to conceptualize at a glance how many possible starting states would evolve over time. Operations like [Divergence](Divergence.md) and [Curl](Curl.md) can help to inform us about the system. Do the population sizes tend to converge towards a particular pair of numbers, or are there some values that they diverge away from. Are there cyclic patterns? Are those cycles stable or unstable? Note that in cases like these we would likely want to analyze the jacobian! --- Links to: [Vector-Fields](Vector-Fields.md) [Divergence](Divergence.md) [Curl](Curl.md) References: * [Divergence and Curl - 3b1b](https://youtu.be/rB83DpBJQsE?t=477)