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-------------------- 2478 - EQUATIONS - making sense without the math.
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- SEE : Review 1865, 5 pages, that contains more math formulas entitled “ Equations are just another language to learn“. This review tries some explanations without the math symbols, ie; the shorthand.
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- GENERAL RELATIVITY:
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- The General Relativity Equation was formulated by Einstein as part of his groundbreaking general theory of relativity in 1915. The theory revolutionized how scientists understood gravity by describing the force as a warping of the fabric of space and time.
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- The right-hand side of this equation describes the energy contents of our universe ,including the 'dark energy' that propels the current cosmic acceleration.
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- The left-hand side describes the geometry of space-time. The equality reflects the fact that in Einstein's general relativity, mass and energy determine the geometry, and the curvature of space, which is a manifestation of what we call gravity.
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- The equation reveals the relationship between space-time and matter and energy. "The presence of the Sun warps space-time so that the Earth moves around it in orbit.
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- The “standard model” in physics describes the collection of fundamental particles currently thought to make up our universe.
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- The theory can be encapsulated in a main equation that successfully describes all elementary particles and forces that we've observed in the laboratory to date, except gravity.
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- The standard model theory has not yet, however, been united with general relativity, which is why it cannot describe gravity.
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- THEOREM FOR CALCULUS:
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- The fundamental theorem of calculus forms the backbone of the mathematical method known as calculus, and links its two main ideas, the concept of the integral and the concept of the derivative.
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- In simple words, it says that the net change of a smooth and continuous quantity, such as a distance traveled, over a given time interval is equal to the integral of the rate of change of that quantity, i.e. the integral of the velocity.
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- The seeds of calculus began in ancient times, but much of it was put together in the 17th century by Isaac Newton, who used calculus to describe the motions of the planets around the sun.
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- PYTHAGOREAN THEOREM:
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- An "oldie but goodie" equation is the famous Pythagorean theorem, which every beginning geometry student learns.
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- This formula describes how, for any right-angled triangle, the square of the length of the hypotenuse, c, (the longest side of a right triangle) equals the sum of the squares of the lengths of the other two sides (a and b). Thus, a^2 + b^2 = c^2
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- INFINITY IN NUMBERS:
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- 1 = 0.999999.….. This simple equation, which states that the quantity 0.999, followed by an infinite string of nines, is equivalent to one. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity."
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- ENERGY AND MASS:
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- Einstein makes the list again with his formulas for special relativity, which describes how time and space aren't absolute concepts, but rather are relative depending on the speed of the observer. His equation shows how time dilates, or slows down, the faster a person is moving in any direction.
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- EULER’S POLYHEDRAL OF SOLID SHAPES:
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- It says that if you cut the surface of a sphere up into faces, edges and vertices, and let F be the number of faces, E the number of edges and V the number of vertices, you will always get V – E + F = 2.
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- For example, take a tetrahedron, consisting of four triangles, six edges and four vertices. If you blew hard into a tetrahedron with flexible faces, you could round it off into a sphere, so in that sense, a sphere can be cut into four faces, six edges and four vertices. And, we see that V – E + F = 2. Same holds for a pyramid with five faces, four triangular, and one square, eight edges and five vertices," and any other combination of faces, edges and vertices.
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- NOETHER’S THEOREM:
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- A spinoff of the Lagrangian equation is called Noether's theorem, after the 20th century German mathematician Emmy Noether. This theorem is really fundamental to physics and the role of symmetry. The theorem is that if your system has a symmetry, then there is a corresponding conservation law.
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- For example, the idea that the fundamental laws of physics are the same today as tomorrow (time symmetry) implies that energy is conserved. The idea that the laws of physics are the same here as they are in outer space implies that momentum is conserved. Symmetry is perhaps the driving concept in fundamental physics, primarily due to Noether's contribution."
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- CALLAN-SYMANIZIK EQUATION:
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- The Callan-Symanzik equation is a vital first-principles equation from 1970, essential for describing how naive expectations will fail in a quantum world. The equation has numerous applications, including allowing physicists to estimate the mass and size of the proton and neutron, which make up the nuclei of atoms.
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- Basic physics tells us that the gravitational force, and the electrical force, between two objects is proportional to the inverse of the distance between them squared. On a simple level, the same is true for the strong nuclear force that binds protons and neutrons together to form the nuclei of atoms, and that binds quarks together to form protons and neutrons.
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- However, tiny quantum fluctuations can slightly alter a force's dependence on distance, which has dramatic consequences for the strong nuclear force. It prevents this force from decreasing at long distances, and causes it to trap quarks and to combine them to form the protons and neutrons of our world.
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- What the Callan-Symanzik equation does is relate this dramatic and difficult-to-calculate effect, important when the distance is roughly the size of a proton, to more subtle but easier-to-calculate effects that can be measured when the distance is much smaller than a proton."
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- The minimal surface equation somehow encodes the beautiful soap films that form on wire boundaries when you dip them in soapy water. The fact that the equation is 'nonlinear,' involving powers and products of derivatives, is the coded mathematical hint for the surprising behavior of soap films.
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- This equation is in contrast with more familiar linear partial differential equations, such as the heat equation, the wave equation, and the Schrödinger equation of quantum physics."
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- See review 1865 for more about math equations and the story they tell. Simplification in formulas, helps us know more and more about less and less until we know everything about nothing.
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- See review 1953 “ Math was invented to solve problems”, some 20 more Reviews about math listed at the end of the Review. This is the end of this Review without the math.
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- November 10, 2019 2479
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--------------------- Monday, November 11, 2019 --------------------
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