Physics Teacher's Notes

. Related post:  [MATPY00] The Exponential Function   Our MATPY00 class collected the data on the confirmed cases of COVID-19 in the Philippines (data is available through this link ). The purpose of this post is to provide a data-based analysis (an "experiment") to apply the things we learned about the exponential function, as detailed in the previous post .  First, we realized that the first infected individual in the Philippines was reported on January 30, 2020. Afterwards, the class collected the data from official bulletins from the Department of Health (DOH) and news reports to reconstruct the time series of the number of infected individuals. The plot of the increase in the number of infections is plotted in Figure 1 below.  Figure 1. The number of COVID-19 infected individuals in the Philippines, plotted with day number (with day 0 corresponding to January 30, 2020). The rapid rise, especially after day 40, is indicative of exponential growt...

. Related post:  [GENPHY2] Right Hand Rule   At this point, let us consider one example of the right hand rule in nature. The magnetic force \(\vec{F}_m\) experienced by a moving charge \(q\) with velocity \(\vec{v}\) in a region with magnetic field \(\vec{B}\) is given by:  \[\begin{equation}\label{eqn:fm} q \vec{v} \times \vec{B} = \vec{F}_m\end{equation}\]  Just as in the previous post , let us use color codes for the vectors involved in the cross product:  The first vector \(q\vec{v}\) is red .  The second vector \(\vec{B}\) is green .  The third vector (the result of the cross product) \(\vec{F}_m\) is blue .  Before we proceed, notice that I specifically included the \(q\) (which is a scalar quantity) in the first vector. This has an important implication. The direction of the first vector is not simply the direction of the velocity; instead, it should include the effect of the sign  of the charge \(q\)....

. The exponential function is a special kind of function because it is the solution to the following differential equation:  \[\begin{equation}\label{eqn:diffeqn}\frac{dx}{dt} = rx\end{equation}\] i.e. the derivative (here shown as a rate of change) in the value of \(x\) is proportional to itself. The constant \(r\) is the exponential rate: if \(r > 0\), we will find an exponential growth, while if \(r < 0\), we will have an exponential decay.  Before solving Eqn (\(\ref{eqn:diffeqn}\)), let us discuss why this equation is important. Many processes in the real world starts out with trends comparable to the one modeled by this equation. For example, consider the case of the spread of infectious diseases, just like the one we are having right now. Let \(x(t)\) represent the number of infected individuals at a particular time \(t\). These infected individuals are the ones who can transmit the infection to the other individuals; more other people can be infecte...

. The right hand rule (RHR) is important for understanding the directions of the vectors involved in a cross product. Let me teach you some techniques and tricks that work. 😊  RHR for the Coordinate Directions  Coordinate Systems are Right-Handed. The three-dimensional coordinate systems that we use in mathematics actually follow the RHR. For example, the rectangular (Cartesian) coordinate system \(xyz\) follows this relationship for their unit vectors:  \[\begin{equation} \label{eqn:rhrxyz} \hat{i} \times \hat{j} = \hat{k}\end{equation}\]  Therefore, in understanding the RHR for cross products, we can refer to the right-handedness of coordinate systems. The Cartesian coordinate system is illustrated in Figure 1 below using the common directions we associate with them: \(+x\) is rightward, \(+y\) is upward, and \(+z\) is outward (by the way, the circle with a dot inside means outward, while the circle with an "X" inside means inward). For our ...