Guess what the world is weird, and when i say weird i mean real weird.
In thermodynamics one of the most interesting findings is the point that we call the triplet point. At the triplet point a given material will be a liquid, gas and solid. The material will quickly like an explosion boil and then in second freeze to alter become a liquid and then begin the loop again. The reaction will uncontrollably shift back and forward. (We really recommend that you check a video out on this, some are really wild)
To understand this phenomena outside of our normal understanding we need to make mathematical models and the mathematical model we use to understand triplet points have to make use of what we call implicit function.
Implicit function is what you get when we can not use normal functions. A normal function maps all to a unique . If this is not true we need implicit functions and for the case of triplet point we have this exact case. Here a specific combination of temperature and pressure gives three possible states liquid, solid and gas.
This specific result can be partly explained with the ideal gas law that states that
Here is pressure, is a volume, and can for our example be regarded as constant and is temperature. The three variables are dependent on each other and is an implicit relation. As you can see if for example is constant then must rise as does, this is one of the clues for why a triplet point exist.
An implicit function is a function that is implicitly defined in an equation, that is, we have to solve for in the equation. An example of this is the equation for the unit circle
defines the unit circle, but only implicitly, since we have to solve for it. For some equations however, it may not even be possible to solve for at all.
Even if we can't solve for in these equations, the implicit function theorem tells us that still defines a function, if some conditions are fulfilled.
Here, have an equation:
This is its graph:
We'll have a rough time if we attempt to disentangle the from the rest to write the equation as an explicit function .
Look at the graph above. To the left of the point indicated on the upper curve, the graph does not represent a function : recall that for it to be a function, there can only be one -value for each input .
However, to the right of the point, the graph looks suspiciously function-like. There, we say that implicitly defines a function . Indeed, even if some equation is not a function on its whole domain, it can often be regarded as one locally.
Example 1: the unit circle
This is the equation of the unit circle:
Trying to isolate the , we get:
We cannot express the unit circle equation as one function on its whole domain. The problem is that for every , there are two -values.
Locally, however, we can almost everywhere interpret the equation as a function , by limiting the domain of to an area around the point of interest.
There are only to points where we cannot define locally: at and . Notice that these are the two points where , and the gradient is horizontal.
A side note: notice that we said the domain of . The domain is made up of pairs of points. Actually, is a level curve of the function in two variables. Thus, its domain consists of pairs of points.
Sometimes, we may instead want to parameterize the curve with as variable. Then, we'd have . All we'd like to do with ta function of can be done with this function of , we just have to be careful to implement this name change into all that we do.
In the unit circle case, we'd be able to define the circle as a function of locally everywhere except at and , instead. There, we have . This is no coincidence, and keeping this in mind will be helpful soon, as we talk about the implicit function theorem.
Example 2: the unit sphere
In higher dimensions, implicit functions function in similar ways.
Let's look at the unit sphere, which is actually a level surface to :
Notice that there are two values for at every point , . However, we can regard the sphere as a function surface locally, if we restrict the set of points in the domain of to lies close to the point we're looking at.
The only places where we cannot let implicitly define a function is around the equator of the sphere. There, .
Suppose you've got some kind of equation, like something even worse . As it turns out, this thing might define a function. At least if you zoom in on a particular interval, where the points make up a function graph. I mean, how neat isn't that? That's an implicit function, right there.
Ok, end of the "math is cool" bit. Time to dive into the details.
Back in single variable calc we got acquainted with the unit circle, . Then we told you to take the derivative of the left side and the right side with respect to , and solve for . That'd give you the slope of the tangent at that point.
But didn't it seem kinda random? Who came up with the idea of just differentiating the left hand side, and then differentiating the right hand side? It doesn't really make sense.
Multivariable calc lends itself to an alternative explanation. You can basically treat the left side as a level curve, . Now assume can be written as a function . Let's go ahead and take the partial derivative with respect to , so
Since it's a level curve, it obviously doesn't change, and so the derivative is . Then you can solve for , getting the same result as before: . And yeah, notice that . Here, we require that .
The method discussed above is easily generalized to a function . We will do this in the following example
Find the following partial derivatives at and , at the point . First we define
Then we set
Therefore we find that
The same argument can be used on
Next we calculate all the partial derivatives
Next we use the point
The implicit function theorem
We'll here gather the information which we've thrown out in the previous notes on implicit functions.
Recall that we said that if is some level curve, then it often implicitly defines a function locally. This works well if and only if the gradient at the point is not parallel to the -axis, or, equivalently, that .
Then, using the chain rule on , we found the derivative for :
Studying that formula, we see that if , has a vertical tangent line. This would mean is not a function of , as there would be several function values for the same .
The above facts constitute the essence of the implicit function theorem:
Let be a differentiable function and a point on the level curve . If
there exists an open domain to such that the restriction of on the level curve implicitly defines a differentiable function . The derivative of this function is:
Note that we may flip the and the around, defining an implicit function . Then, the requirement is .
There exist versions of this theorem for higher dimensions as well. In particular, the condition
is sufficient for us to be able to regard the level surface as a function surface
The geometrical interpretation of this is that if , the tangent plane to at the point isn't perpendicular to the -plane. This is exactly what we need for a function surface .
The partial derivatives of the implicit function are:
Like in the first version of the theorem, we can exchange the variables. For example, is a sufficient criterion for being able to regard as a function around .
Use the implicit function theorem to show that we cannot assume that the curve can be parameterized, using at
To prove the statement above we can use the implicit function theorem. The implicit function theorem works on all variables in the implicit expression . We just have to make sure to check the right partial derivative of . Therefore, we calculate
Using our point we see that
Therefore we can not be certain that curve can be parameterized around the point.