Intro
Albert Einstein wrote in the year in 1935 in an article in the New York Times, after having heard his friend Emmy Noether had died. In that article the following quote can be found.
Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.
Noether described how each symmetry in a field that exists will result in one conservation law. For example the fact that we have symmetry in time is why we have energy conservation. This idea is one of the foundations of mathematics physics today.
Emmy was born a Jew during the Nazi era in Germany and much of her work was criticized solely on the fact of her gender and ethnicity. But her works did not go to waste and her academic contribution to the fields of math and physics gets more and more of the recognition it deserves.
Concept
If you assign a vector for each point in the -plane, you get a vector field. Vector fields are useful in describing phenomenon that has a direction and magnitude.
The wind flow is a good example, since it has different direction och strength depending on where you are in the world.
Similarly, a scalar field assigns a number to each point in the -plane. Temperature is a good example of a scalar field since it varies depending on where you are in the world.
Math
As we have already seen there are a lot of application of fields. But to truly understand field we need to look how they work on dimensions. We can first observe this vector field:
The field can be expressed with the following function
Note that this function is vector and we can therefore write that
Looking at the scalar function in the previous slide we have a function on the shape
Note that this function is a scalar, and therefore we find that
Vector fields
This is a vector field:
The arrows' direction and length indicate the direction and magnitude of the field at that point.
For instance, vector fields can describe the water flow at the surface of a bathtub. At each point, the vector field tells us how quickly and towards where the water flows.
If you'd put a little rubber duck on the waves, the duck would follow the streams at the surface as it wobbles around; it would follow the arrows.
Recall that we talked about vector valued functions
These take in a scalar value and spits out a vector. Specifically, we used them for parameterizing curves.
As we were parameterizing surfaces, we actually dealt with vector valued functions in two variables, or functions
They took in two coordinates and gave us three, the coordinates of the surface.
Now, a vector field is a function
It associates with each point in the plane a vector. This vector can tell us the magnitude and the direction of the field at the point.
There are also vector fields
For example, pulling the plug in the bathtub gives rise to a vector field in three dimensions at the water surface, describing how quickly the water flows.
A vector field in three dimensions can be written like this:
where , and are the unit vectors in .
Note that the subscripts indicate that these are vector components, not to be confused with partial derivatives.
In two dimension, we can write the field in this way:
Vector fields we've met
We've already seen some vector fields, though we haven't explicitly called them that.
For example, the gradient field associates with each point on a function surface a vector, telling us how quickly and towards where the function grows at the point.
The gravitational field may also ring a bell. It is what keeps the Earth around the Sun. The field of gravity points in towards the Sun, and the magnitude is greatest at the surface of the Sun itself.
Example
Look at the figure below.
That is a plot of the vector field
It also can be written as
or
Scalar fields
As you come home from school, you suddenly feel like drawing a map of your room. Who cares about YouTube and social media if you can draw maps of your room? You'll draw your room from above, giving sort of a bird's eye view.
In your room, the temperature varies. It's slightly colder near the window, and it's warmer near the radiator. You could, if you're hard core, color your map blue and red to indicate the color.
That's how you'd visualize a scalar field. If you've got a function which associates every point in the plane with some value, like the temperature, you can plot it as a scalar field. It's essentially just another way of thinking about "ordinary" multivariable functions. Rather than drawing a graph with the values, you can color your plane.
Example
Have a look at the function . If you'd draw it the 'standard way', you'd get something like this:
But you could also represent it as a scalar field. Here's what you'd get:
Field lines
Remember Ducky, the rubber duck in our bath tub? We could trace out the path along which Ducky travels. The path evidently depends on the vector field, but also on where we place Ducky.
If we place him at a given point, he might be caught in a current and drift off towards the left. Had he had a different starting position, he might have ended up elsewhere.
And if you'd also sketch the vector field on top, you'd get this kind of pic:
The snake-like line is what's known as a field line. Notice how the field line is tangent to the vectors at every point. The field lines don't tell us anything about the magnitude, but it gives us the direction at every point.
