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Kurs: Analiza matematyczna funkcji wielu zmiennych > Rozdział 2
Lekcja 10: Dywergencja i rotacja (artykuły)Curl warmup, fluid rotation in two dimensions
Curl measures the rotation in a fluid flowing along a vector field. Formally, curl only applies to three dimensions, but here we cover the concept in two dimensions to warmup.
Kontekst
Note: Throughout this article I will use the following convention:
represents the unit vector in the -direction. represents the unit vector in the -direction.
Do czego zmierzamy
- Curl measures the "rotation" in a vector field.
- In two dimensions, if a vector field is given by a function
, this rotation is given by the formula
Rotation in fluid flow
Have yourself a nice swirly vector field:
This particular vector field is defined with the following function:
Now I want you to imagine that this vector field describes a fluid flow, perhaps in a chaotic part of a river. The following video shows a simulation of what this might look like. A sample of fluid particles, shown as blue dots, will flow along the vector field. This means that at any given moment, each dot moves along the arrow it is closest to. Focus in particular on what happens in the four circled regions.
Amidst all the chaos, you might notice that the fluid is rotating within the circled regions. In the left and right circles, the rotation is counterclockwise, and in the top and bottom circles, the rotation is clockwise.
- Key Question: If we are given a function
that defines a vector field, along with some specific point in space, , how much does a fluid flowing along the vector field rotate at the point ?
The vector calculus operation curl answer this question by turning this idea of fluid rotation into a formula. It is an operator which takes in a function defining a vector field and spits out a function that describes the fluid rotation given by that vector field at each point.
Technically, the curl operation only applies to three dimensions. You can see what that means and how it is computed in the next article, but in this article, we warm up by describing fluid rotation in two dimensions with a formula.
Capturing two-dimensional rotation with a formula
One of the simplest examples of a vector field which describes a rotating fluid is
Here's what it looks like.
Animated, all the fluid particles just go in circles.
In some sense, this is the most perfect example of counterclockwise rotation, and you can understand the general formula for rotation in a two-dimensional vector field just by understanding why the function gives counterclockwise rotation.
The -component
First, let's understand why the component suggests counterclockwise rotation. Imagine a small twig sitting in our fluid, oriented parallel to the -axis. More specifically, let's say one end is at the origin , and the other is at the point . What does the component of the vector field imply for the fluid velocity at points on this twig?
This means the velocity at the top of the twig is , a leftward vector, while the velocity at the bottom of the twig is .
For the twig, this means the important factor for counterclockwise rotation is that vectors point more to the left as we move up the vector field. Said with a few more symbols, the important point here is that the -component of a vector attached to a point decreases as increases.
Said with even more symbols,
Let's generalize this idea a bit.
- Question: Consider a more general vector field.
The components and are any scalar-valued functions. If you place a small twig at some point , oriented parallel to the -axis, how can you tell if the twig will rotate just by looking at and ?
- Answer: Look at the rate of change of
as varies near the point of interest, :If this is negative, it indicates that vectors point more to the left as increases, so rotation would be counterclockwise. If it is positive, vectors point more to the right as increases, indicating a clockwise rotation.
The -component
Next, let's see why the component of the original vector field suggests counterclockwise rotation as well. This time, imagine a twig which is parallel to the -axis. Specifically, put one end of the twig at the origin , and put the other at the point .
The vector attached to the origin is , but the vector attached to the other end at is , an upward vector. Therefore, the fluid pushes the right end of the stick upwards, and the left end experiences no force, so there will be a counterclockwise rotation.
For this second twig, the vertical component of vectors increases as we move right, suggesting counterclockwise rotation. That is to say, the component of a vector attached to a point increases as increases.
In the case of a more general vector field function,
we can measure this effect near a point by looking at the change in as changes.
Combining both components
Putting these two components together, the rotation of a fluid flowing along a vector field near a point can be measured using the following quantity:
When you evaluate this, a positive number will indicate a general tendency to rotate counterclockwise around , a negative quantity indicates the opposite, clockwise rotation. If it equals , there is no rotation in the fluid around . If you are curious about the specifics, this formula gives precisely twice the angular velocity of the fluid near .
Some authors will call this the "two-dimensional curl" of . This isn't standard, but let's write this formula as if "2d-curl" was an operator.
Example: Analyzing rotation in a 2d vector field using curl
Problem: Consider the vector field defined by the function
Determine whether a fluid flowing according to this vector field has clockwise or counterclockwise rotation at the point
Krok 1: Compute the of this function.
Krok 2: Plug in the point .
Krok 3: Interpret. How does the fluid tend to rotate near this point?
Let's watch a sample of particles in this fluid flow:
The point towards the top where all the particles congregate corresponds with . Particles rotate counterclockwise in this region, which should be consistent with your calculations.
Podsumowanie
- Curl measures the "rotation" in a vector field.
- In two dimensions, if a vector field is given by a function
, this rotation is given by the formula
On to the third dimension!
The true curl operation, covered in the next article, extends this idea and this formula to three dimensions.
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