## Tuesday, April 6, 2010

### Mathematics Guide

"THIS is a football ..."
... Vince Lombardi

 The language of physics is mathematics. In order to study physics seriously, one needs to learn mathematics that took generations of brilliant people centuries to work out. Algebra, for example, was cutting-edge mathematics when it was being developed in Baghdad in the 9th century. But today it's just the first step along the journey. Algebra Algebra provides the first exposure to the use of variables and constants, and experience manipulating and solving linear equations of the form y = ax + b and quadratic equations of the form y = ax2+bx+c. Geometry Geometry at this level is two-dimensional Euclidean geometry, Courses focus on learning to reason geometrically, to use concepts like symmetry, similarity and congruence, to understand the properties of geometric shapes in a flat, two-dimensional space. Trigonometry Trigonometry begins with the study of right triangles and the Pythagorean theorem. The trigonometric functions sin, cos, tan and their inverses are introduced and clever identities between them are explored. Calculus (single variable) Calculus begins with the definition of an abstract functions of a single variable, and introduces the ordinary derivative of that function as the tangent to that curve at a given point along the curve. Integration is derived from looking at the area under a curve,which is then shown to be the inverse of differentiation. Calculus (multivariable) Multivariable calculus introduces functions of several variables f(x,y,z...), and students learn to take partial and total derivatives. The ideas of directional derivative, integration along a path and integration over a surface are developed in two and three dimensional Euclidean space. Analytic Geometry Analytic geometry is the marriage of algebra with geometry. Geometric objects such as conic sections, planes and spheres are studied by the means of algebraic equations. Vectors in Cartesian, polar and spherical coordinates are introduced. Linear Algebra In linear algebra, students learn to solve systems of linear equations of the form ai1 x1 + ai2 x2 + ... + ain xn = ci and express them in terms of matrices and vectors. The properties of abstract matrices, such as inverse, determinant, characteristic equation, and of certain types of matrices, such as symmetric, antisymmetric, unitary or Hermitian, are explored. Ordinary Differential Equations This is where the physics begins! Much of physics is about deriving and solving differential equations. The most important differential equation to learn, and the one most studied in undergraduate physics, is the harmonic oscillator equation, ax'' + bx' + cx = f(t), where x' means the time derivative of x(t). Partial Differential Equations For doing physics in more than one dimension, it becomes necessary to use partial derivatives and hence partial differential equations. The first partial differential equations students learn are the linear, separable ones that were derived and solved in the 18th and 19th centuries by people like Laplace, Green, Fourier, Legendre, and Bessel. Methods of approximation Most of the problems in physics can't be solved exactly in closed form. Therefore we have to learn technology for making clever approximations, such as power series expansions, saddle point integration, and small (or large) perturbations. Probability and statistics Probability became of major importance in physics when quantum mechanics entered the scene. A course on probability begins by studying coin flips, and the counting of distinguishable vs. indistinguishable objects. The concepts of mean and variance are developed and applied in the cases of Poisson and Gaussian statistics.

#### 2 comments:

Steven Colyer said...

"After serving as a member of the Revolutionary Committee that terrorized France, sent Coulomb into hiding, arrested Lagrange and guillotined Lavoisier, a repentant Jean Baptiste Joseph Fourier causes controversy with his memoir On the Propagation of Heat in Solid Bodies. His former teachers Laplace and Lagrange object to his use of infinite trigonometric series, which we now call Fourier series. Fourier later wins the Paris Institute Mathematics Prize for solving the problem of heat propagation, over the repeated objections of Laplace and Lagrange."

Steven Colyer said...

Charming. Then there's this:

1615: Kepler's mother, Frau Katharina Kepler, is accused of witchcraft by a local prostitute. European witch hunting was at its peak during Kepler's career, and witch hunting was supported by all levels of society, including secular officials and intellectuals in universities. Kepler spends the next several years making legal appeals and hiding his mother from legal authorities seeking to torture her into confessing to witchcraft. Examining an accused witch ad torturam was a standard court procedure during this era.

1619: Under court order, Kepler's mother is kidnapped in the middle of the night from her daughter's home and taken to prison. Kepler spends the next year appealing to the duke of Württemberg to prevent his imprisoned mother from being examined ad torturam.

1621: On September 28, Katharina Kepler is taken from her prison cell into the torture room, shown the instruments of torture and ordered to confess. She replies "Do with me what you want. Even if you were to pull one vein after another out of my body, I would have nothing to admit," and says the Lord's Prayer. She is taken back to prison. She is freed on October 4 upon order of the duke, who rules that her refusal to confess under threat of torture proves her innocence. He also orders her accusers to pay the cost of her trial and imprisonment.

1622: After having spent most of the last seven years under the legal threat of imminent torture, Katharina Kepler dies on April 13, still being threatened with violence from those who insist she is a witch.

And they said mathematics was boring. :-)