Marvelous Mathematics and Fantastic Physics

Mazes are a very accurate representation of the process that goes behind solving differential equations. When initially experimenting with them, without any prior knowledge, they may seem impossible. However, when provided with a bird's eye view of the layout, the process suddenly seems simple. And with practice, it becomes easier and quicker to reach the final destination. This description is most applicable for first order, linear differential equations , a set of differential equations of the simplest kind.

Verification of a differential equation is exactly like plugging in the last piece of a jigsaw puzzle; the solution is inserted back into the differential equation to ensure that it qualifies its conditions . Apart from the satisfaction of finding out a successful solution that may have taken hours of effort, learning to verify accurately is also important for various other reasons. Fortunately, verification is far simpler than actually solving most differential equations.

Differential equations exhibit fascinating variation , with a division as diverse as regular, algebraic equations. Possessing knowledge about the various types and correctly applying that knowledge to identify a particular one is a key step in solving any differential equation. This is because, it gives an idea/hint about which method can be used in that particular scenario, saving time and energy. While the classification can get very complex, most of the basic, yet crucial, differences are highlighted here.

Differential Equations hold a very special place in the field of mathematics, forming the strong interlocking chain that bridges the gap between abstract mathematics and each of the practical sciences . Principally, differential equations are a subset of calculus, as every differential equation works with functions and their derivatives . Much like algebraic equations, some differential equations are easy enough to be solved by any novice, while others have remain unsolved and teased centuries of geniuses.

While the constancy over the speed of light in a vacuum is a fundamental law that cannot be challenged (in fact it was one of the postulates of special relativity), the wavelength of light is definitely a factor that can vary. In general relativity, Einstein highlights the consequence of gravitational redshift, a phenomenon where light emitted from a huge mass changes its wavelength when escaping the mass' gravitational field .

Do you feel dizzy looking at the clock above? Equally fascinating and perplexing as the picture above, is general relativity's consequence of gravitational time dilation. While this concept was introduced in special relativity as a result of travelling at relativistic speeds, its application in general relativity's gravitation has more prominent, observable examples. This concept, now confirmed experimentally, describes how time runs slower in a stronger gravitational field .

         Gravitational lensing is truly one of the most stunning applications of general relativity, as seen with the ring in the picture above. Apart from its physical beauty, the treasure lies in the knowledge of how the ring is formed in the first place. While it may seem like highly-condensed matter at first, it is actually just the light from stars bent around the massive central object . As counter-intuitive as that may seem, if spacetime curves due to massive objects, so does the path of light in it.

General relativity is one of the most notable theories in physics, with the potential to explain most of the phenomena we encounter in our universe. In fact, general relativity has even been part of the theory scientists have been yearning for: the theory of everything (TOE) . Unlike special relativity, which can be understood with high-school math, general relativity can get very complicated with its differential equations .

As humans, we have the tendency to explore events in greater depth and detail. To be more specific, we strive to find the causes and consequences for all events in the universe. However, in the scientific world, there are some restrictions for these answers. These were derived from Einstein's special relativity which highlighted how the speed of light in a vacuum was the universal speed limit . For the sake of simplicity, this concept is typically illustrated with a light cone .

                                                                                                                                                                As we scramble for efficiency in the 21st century, one idea is evident: the need for speed is critical. While technological advancements have definitely improved how quickly tasks are done, the question that arises is what is the universal speed limit for everything? Fortunately, special relativity provides us with a fitting answer to this conundrum, using the most famous equation ever written.

                                                                                                                                                                                 One of the biggest myths of all time is that time runs at the same rate for everyone. Einstein's theory of special relativity completely disproved this statement by showing how one of the most prominent effects of travelling at high, relativistic speeds is time dilation. This concept, now confirmed experimentally, describes how time runs slower at higher velocities.

                                                    Do you notice the wheels turning in this picture? Much like the picture, special relativity gets confusing because of its counterintuitive use of perspectives . As discussed in the previous post, special relativity is based upon 2 deceivingly clever postulates. By solely using these 2 ground rules, a variety of implications, connecting to scenarios of travelling at relativistic speeds , can be traced out. 

          Special relativity is one of the most widely known theories in physics, being the solid foundation or bedrock for physics that drastically revolutionizes our understanding of the universe. While special relativity does not require more than basic high-school knowledge, a majority of people fail to fully grasp the intricacies of special relativity.

Motion in air is a prime example of how  physics and mathematics dominates our lives, exemplifying their synergy. Most physics laws and calculations need complex mathematics to be proven/justified, but mathematics by itself isn't particularly useful in terms of its application in the real world. Thus, this page would mainly show how in-flight motion employs the need to use both physics and mathematics together for better understanding.