Many readers know that reflecting telescopes, radio telescopes, and solar collectors all feature a parabolic surface. These devices utilize the property of parabolic surfaces to converge parallel incoming light onto a single point (the focus). If asked why a parabola possesses this property, I believe many high school students could prove it using their knowledge of parabolas. However, if the question is reversed: Why must the surface that possesses this property be a parabola? I believe this might stump some readers. Let us attempt to find this curve (due to symmetry, this surface can be viewed as a surface of revolution generated by a curve; thus, we can study the curve itself).
Construct a reflection diagram as shown above. Define a constant unit vector \vec{C} (fixed direction, |\vec{C}|=1) to represent the direction of the parallel incoming light. Construct a vector \vec{r} starting from the focal point to represent the trajectory of the curve, and let |\vec{r}|=r. According to the law of reflection (angle of incidence = angle of reflection), the angles between d\vec{r} and the vectors \vec{r} and \vec{C} must be equal. Thus, we have:
\vec{C} \cdot d\vec{r} = \frac{\vec{r} \cdot d\vec{r}}{r} (This can be derived by considering the meaning of both sides of the equation in connection with the definition and calculation of the vector dot product.)
We have the identity: \vec{r} \cdot d\vec{r} = r dr (note that dr is d(|\vec{r}|) and not |d\vec{r}|). Substituting this in, we immediately obtain:
\vec{C} \cdot d\vec{r} = dr
Integrating once, we get:
\vec{C} \cdot \vec{r} = r + K
Where K is a constant. If we convert this back to the Cartesian coordinate system or polar coordinate system, we will find that the above equation represents a parabola. Thus, the problem is solved.
In the discussion above, we used only a few steps, which fully demonstrates the great advantage of using vectors. Initially, BoJone tried to list the equation directly in Cartesian coordinates as dy\sqrt{x^2+y^2} = xdx + ydy, and then transformed it into polar coordinates, which was particularly cumbersome.
This problem was proposed a year ago, originating from one of the ten thought experiments Einstein posed to young people: “A solar furnace that does not move with the sun.” BoJone provided the solution on November 3, 2010. This problem can be considered the first differential equation BoJone ever formulated. At that time, let alone solving it, even listing the equation took a lot of effort. Having just encountered calculus and differential equations and not yet having systematic exposure to knowledge like vectors, I relied solely on weak analytical geometry and various brute-force methods to list the equation, which was eventually solved with the help of a teacher. Additionally, the “Cow-Chasing Problem” mentioned in a previous article was also one of the early practical differential equation problems BoJone encountered. These initial encounters with such problems led to my longing for in-depth mathematical research and pushed me to study mathematics.