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Confirming Einstein’s Theory of General Relativity With Calculus, Part 6e: Rationale for Method of Undetermined Coefficients II – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. We…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 6d: Rationale for Method of Undetermined Coefficeints I – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. We…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 6c: Solving New Differential Equation with Variation of Parameters – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. Under…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 6b: Checking Solution of New Differential Equation with Calculus – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. In…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 6a: New Differential Equation – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. We…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 5d: Deriving Orbits under Newtonian Mechanics Using Variation of Parameters – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. We…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 5c: Deriving Orbits under Newtonian Mechanics with the Method of Undetermined Coefficients – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. We…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 5b: Deriving Orbits under Newtonian Mechanics with Calculus – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. We…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 5a: Confirming Orbits under Newtonian Mechanics with Calculus – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. We…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 4c: Newton’s Second Law and Newton’s Law of Gravitation – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. In…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 4b: Acceleration in Polar Coordinates – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. In…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 4a: Angular Momentum – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. In…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 3: Method of Successive Approximations – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. One…
A Postcard from Spokane – Mean Green Math
A brief aside from the current series on general relativity --- and the mysterious 43 seconds of arc per century in Mercury's orbit --- that turned into further discussion about angle measurement. A few months ago, I received this clever postcard from someone visiting Spokane, Washington. The sender clearly knew the recipient (me) well: rather…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 2d: Hyperbolas and Polar Coordinates – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. In…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 2c: Circles, Parabolas, and Polar Coordinates – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. In…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 2b: Ellipses and Polar Coordinates – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. As…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 2a: Graphically Exploring Precession – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. But…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 1c: Outline of Argument – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. This…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 1b: Precession of Mercury – Mean Green Math
In this series, I'm discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury's orbit and thus confirm Einstein's theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago. The…
Confirming Einstein’s Theory of General Relativity With Calculus, Part 1a: Introduction – Mean Green Math
If the universe consisted of only Mercury and the sun, Mercury's trajectory would trace the same ellipse over and over again. However, there are seven other planets in the solar system (not to mention the dwarf planets), and these planets tug and nudge the orbit of Mercury ever so slightly. (For what it's worth, similar…
The Rocket Scientist Quarterbacking an Unlikely NFL Playoff Contender – Mean Green Math
I greatly enjoyed this Wall Street Journal article about Joshua Dobbs, currently the quarterback of the Minnesota Vikings. The opening paragraphs: When quarterback Joshua Dobbs subbed in for the Minnesota Vikings last week and led them to a dramatic victory just days after they traded for him, it amazed his teammates whose names he barely…
Snell’s Law and a mystery novel – Mean Green Math
Lately, for my own leisure reading, I've been enjoying the murder-mystery novels of Dorothy Sayers. Her books are an enjoyable trip back in time, as she paints a very vivid portrait of English life of during the interwar years of the 1920s and 1930s. (Of course, at the time she was writing, no one had…
Parabolas from String Art: Index – Mean Green Math
I'm doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on numerical integration. Part 1: Introduction Part 2: Identifying the highest points of the strings Part 3: These nine points lie on a parabola: Method #1 Part 4:…
An algebra and trigonometry–based proof of Kepler’s First Law – Mean Green Math
The proofs of Kepler's Three Laws are usually included in textbooks for multivariable calculus. So I was very intrigued when I saw, in the Media Reviews of College Mathematics Journal, that somebody had published a proof of Kepler's First Law that only uses algebra and trigonometry. Let me quote from the review: Kepler’s first law…
Square roots and Logarithms Without a Calculator: Index – Mean Green Math
I'm doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on computing square roots and logarithms without a calculator (with the latest post added). Part 1: Method #1: Trial and error. Part 2: Method #2: An algorithm…
Abraham Lincoln and Geometry – Mean Green Math
While re-reading the wonderful parallel biography Team of Rivals: The Political Genius of Abraham Lincoln by Doris Kearns Goodwin, I was reminded of this passage from Lincoln's time on the Illinois traveling law circuit in the 1850s, the interlude between his term in the House of Representatives and his ascent to the presidency: Life on…
Predicate Logic and Popular Culture (Part 275): Florida Georgia Line – Mean Green Math
Let $latex P$ be the set of all people, and let $latex f(x)$ be the amount that $latex x$ loves you. Translate the logical statement $latex \forall x \in P(f(x) \le f(\hbox{God}) \land f(x) \le f(\hbox{your mama}) \land f(x) \le f(\hbox{I}))$. This matches the chorus of the crossover hit "God, Your Mama, and Me" by…
Predicate Logic and Popular Culture (Part 274): George Jones – Mean Green Math
Let $latex T$ be the set of all times, and let $latex L(t)$ be the statement "He loves her at time $latex t$. Translate the logical statement $latex \forall t \in T(((t < 0) \Longrightarrow L(t)) \land ((t \ge 0) \Longrightarrow \sim L(t)))$, where time $latex 0$ is today. Of course, this matches the quintessential…
Predicate Logic and Popular Culture (Part 273): Beauty and the Beast – Mean Green Math
Let $latex P$ be the set of all people, let $latex S(x)$ be the statement "$latex x$ is slick as Gaston," let $latex Q(x)$ be the statement "$latex x$ is quick as Gaston," and let $latex N(x)$ be the statement "$latex x$'s neck is as thick as Gaston's neck." Translate the logical statement $latex \forall…
Predicate Logic and Popular Culture (Part 272): Beauty and the Beast – Mean Green Math
Let $latex T$ be the set of all things, let $latex D(x)$ be the statement "$latex x$ is a dinner," let $latex F(x)$ be the statement "$latex x$ is in France," and let $latex S(x)$ be the statement "$latex x$ is second-best." Translate the logical statement $latex \forall x in T (D(x) \land F(x) \Longrightarrow…