Partition Limits
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First, a word of caution. We are (mostly) sure that the term "partition limit" is not used by anyone else but us. We don't take credit for the term because it is such an obvious term to use. In fact, it is so obvious that we are surprised that it is not widely used. So if you go looking for it somewhere else then don't be surprised if you don't find it. However, the concept is accurate and widely used. In short, we use the term "partition limit" to mean the <Sign in to see all the formulas> logic we describe below. This logic is in most advanced texts in Calculus or Real Analysis. But again, the term "partition limit" probably won't be found on Google or in a book's index.
Second, partition limits are important for developing the logical framework of Integration and defining integrals as Riemann sums. If you are in a Calculus course that is not for mathematicians, but is more like a business Calculus course, then we recommend stopping here and going on to the Fundamental Theorem of Integral Calculus. If you are a mathematics, engineering, or physics major we suggest that you go on.
Partition limits are extremely similar to, but not exactly like function limits. If you are already comfortable with function limits, then you will become comfortable with partition limits very quickly. Let's get started.
In Calculus we deal a lot with function limits where
<Sign in to see all the formulas>
is rigorously defined as: for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas>, then <Sign in to see all the formulas>. In this lesson we want to define things like<Sign in to see all the formulas>
and<Sign in to see all the formulas>
where $L(P)$ and $U(P)$ are the upper and lower sums used to define <Sign in to see all the formulas>.Before we define the partition limit we need to define the norm of the partition. We denote it by <Sign in to see all the formulas> and it is defined as
<Sign in to see all the formulas>
So the norm of the partition is the length of the largest interval of the partition. So now let's define a partition limit.
Partition Limit |
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The partition limit<Sign in to see all the formulas>means that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that for every partition where <Sign in to see all the formulas>, then <Sign in to see all the formulas>. $F(P)$ can be things like $U(P)$, $L(P)$, <Sign in to see all the formulas> or Riemann Sums. |
Before we go on let's make sure we don't confuse $L(P)$ with $L$. The first is a lower sum on an integral and the second is just a number that the partition limit equals.
There are two important things to note about partition limits. The first is that they are conceptual the same thing as function limits. As we squeeze something we get closer to something else. For function limits when our distance from $a$ gets smaller and smaller the function gets closer and closer to $L$. For partition limits, when our partition norm gets smaller and smaller (by adding more points to the partition), $F(P)$ gets closer and closer to the number $L$.
Second, partition limits are not function limits. The easiest way to see this is to note that both $U(P)$ and $L(P)$ are not functions of <Sign in to see all the formulas>. For example, if we have two partitions $P_1$ and $P_2$ whose norms are equal, then <Sign in to see all the formulas> can be a different number from <Sign in to see all the formulas>. So knowing the value of <Sign in to see all the formulas> does not tell us what the value of $U(P)$ is.