We may interpret partial differentiation in the following manner:
The derivative \(f_x(x,y)\) is obtained by applying \(\frac{\partial}{\partial x}\) to the function \(f(x,y)\) from the left.
\(\frac{\partial}{\partial x}\) is neither a number nor a function. It's something different that we call a (partial) differential operator. The same argument applies to \(\frac{\partial}{\partial y}\). This interpretation of differential operators turns out to be useful in many situations. For example, for any constants \(a, b\in \mathbb{R}\), we may consider the following operator \(D\):
If we apply this operator to the function \(f(x,y)\) from the left, we obtain \(af_x(x,y) + bf_y(x,y)\) as a result. That is,
\[Df(x,y) = af_x(x,y) + bf_y(x,y).\]
Note that the result is different if we apply the operator to \(f(x,y)\) from the right, which will be another differential operator rather than a function:
The result is a vector function composed of the partial derivatives of \(f(x,y)\).
The nabla may be regarded as a vector. Suppose we have a vector function \(\mathbf{u}(x,y) = (u(x,y), v(x,y))\). Then, we can take their dot (scalar) product:
Example. For \(a,b\in\mathbb{R}\), let \(a\frac{\partial}{\partial x} + b\frac{\partial}{\partial y}\) be a differential operator that operates on functions of class \(C^{\infty}\). Then, \(\frac{\partial^2}{\partial x\partial y} = \frac{\partial^2}{\partial y\partial x}\). Accordingly, the following holds:
Defining the birth process Consider a colony of bacteria that never dies. We study the following process known as the birth process , also known as the Yule process . The colony starts with \(n_0\) cells at time \(t = 0\). Assume that the probability that any individual cell divides in the time interval \((t, t + \delta t)\) is proportional to \(\delta t\) for small \(\delta t\). Further assume that each cell division is independent of others. Let \(\lambda\) be the birth rate. The probability of a cell division for a population of \(n\) cells during \(\delta t\) is \(\lambda n \delta t\). We assume that the probability that two or more births take place in the time interval \(\delta t\) is \(o(\delta t)\). That is, it can be ignored. Consequently, the probability that no cell divides during \(\delta t\) is \(1 - \lambda n \delta t - o(\delta t)\). Note that this process is an example of the Markov chain with states \({n_0}, {n_0 + 1}, {n_0 + 2}...
Generational growth Consider the following scenario (see the figure below): A single individual (cell, organism, etc.) produces \(j (= 0, 1, 2, \cdots)\) descendants with probability \(p_j\), independently of other individuals. The probability of this reproduction, \(\{p_j\}\), is known. That individual produces no further descendants after the first (if any) reproduction. These descendants each produce further descendants at the next subsequent time with the same probabilities. This process carries on, creating successive generations. Figure 1. An example of the branching process. Let \(X_n\) be the random variable representing the population size (number of individuals) of generation \(n\). In the above figure, we have \(X_0 = 1\), \(X_1=4\), \(X_2 = 7\), \(X_3=12\), \(X_4 = 9.\) We shall assume \(X_0 = 1\) as the initial condition. Ideally, our goal would be to find how the population size grows through generations, that is, to find the probability \(\Pr(X_n = k)\) for e...
In mathematics, we must prove (almost) everything and the proofs must be done logically and rigorously. Therefore, we need some understanding of basic logic. Here, I will informally explain some rudimentary formal logic. Definitions (Proposition): A proposition is a statement that is either true or false. "True" and "false" are called the truth values, and are often denoted \(\top\) and \(\bot\). Here is an example. "Dr. Akira teaches at UBD." is a statement that is either true or false (we understand the existence of Dr. Akira and UBD), hence a proposition. The following statement is also a proposition, although we don't know if it's true or false (yet): Any even number greater than or equal to 4 is equal to a sum of two primes. See also: Goldbach's conjecture Next, we define several operations on propositions. Note that propositions combined with these operations are again propositions. (Conjunction, logical "and"): Let \(P\)...
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