Twierdzenie Ostrogradskiego-Gaussa 2D
Do czego zmierzamy
- The 2D divergence theorem is to divergence what Green's theorem is to curl. It relates the divergence of a vector field within a region to the flux of that vector field through the boundary of the region.
- is a two-dimensional vector field.
- is some region in the -plane.
- is the boundary of .
- is a function which gives outward-facing unit normal vectors to .
- The 2D divergence theorem says that the flux of through the boundary curve is the same as the double integral of over the full region .
- The intuition here is that if represents a fluid flow, the total outward flow rate from , as measured by the flux integral, equals the sum over all the little bits of outward flow at each point, as measured by divergence.
- Often the component functions of are given as and :In this case, once you write both integrals in terms of and , the 2D divergence theorem looks like this:
- Written in this form, it's easier to see that the 2D divergence theorem is secretly just saying the same thing as Green's theorem.
Intuition: Connecting two outward flow measures
The global view: Flux
The local view: Divergence
Want a deeper understanding?
Proof: Flux integrals + Unit normal vector + Green's theorem
Using the 2D divergence theorem?
- Conceptual benefit: It's a great way to deepen your understanding of flux, divergence, and Green's theorem.
- Strategic benefit: Sometimes an example where Green's theorem is used lends itself more naturally to a divergence-based description. For example, if the line integral you want to compute begins its life as a flux integral, rather than expanding out this line integral to make it look like and applying Green's theorem, you could recognize immediately that it's the same as doubly integrating divergence.
- The 2D divergence theorem relates two-dimensional flux and the double integral of divergence through a region.
- Often the vector field is expressed component-wise:In this case, here's how the 2D divergence theorem looks:
- In this form, it is easier to see that the 2D divergence theorem really just states the same thing as Green's theorem.