Five minutes


Can I summarize each course that I took last year in about five minutes? Let's see.

Classical Mechanics:
- Principle of Least Action is equivalent to Newton's Laws.
- Principle of Least Action goes like this: there is a quantity called "Action", which is a function of a trajectory(real or hypothetical). Only trajectories with an extremum action are physically possible.
- If you have N degrees of freedom, you can use this principle to write N equations of motion. Just plug the Lagrangian into Euler's equation.
- If there are constraint forces, find an excuse to put the constraint into the Langrangian (by adding "zero" times a Langrange multiplier). The magnitude of the Lagrange multiplier give you an idea of how "strong" the constraint force is.
- You can rewrite the Lagrangian in another form. Applying the Legendre transform gives the Hamiltonian. The point of doing this is to make it easier to express equations of motion in terms of momentum. You also get first order differential equations instead of second order ones. (Doesn't make it easier to solve though, since they are typically coupled for non-trivial cases)
- Noether's theorem - the coolest thing of all. "Every symmetry has a corresponding conservation law." It sounds like something that came out of Dao De Jing (ok maybe not that much). E.g., if shifting your setup in the x-direction isn't going to affect its motion, then this means that x-momentum is conserved. So this theorem links every displacement with some kind of momentum. BUT! The weirdest thing is, time is associated with Energy! Really interesting, and this will become significant later.
- Central force motion and Scattering. Some mathematical tricks, many useful formulae that you probably shouldn't try to derive by yourself during a midterm. Landau is brilliant, read each page many times.
- Oscillations with driving forces, damping forces and coupling. Driving forces - use Fourier transformation for periodic driving forces, use Green's function for the general case, Laplace transform works well too. Damping forces - memorize/refer to solutions of this kind of second order ODE, and be comfortable with complex exponents, trig functions and hyperbolic trig functions. Coupling - make small oscillation approximations, dump it into a matrix and solve the eigenvalue/eigenfunction problem. Maple/Mathematica is your friend. If coupling is weak, approximate generously.
- Non-intertial frames and rigid body motion. Nightmare fuel. Remember Euler's equations, remember how to use them. Developing an intuition, while not wholly rigorous, helps to prevent sign errors and saves time. There's also a magical equation, "d/dt|_inertial = d/dt|_body - omega cross", which nobody seems to know how to derive but somehow works marvelously well.
- Special relativity. The relativistic Lagrangian is -gamma mc^2. You can obtain it by doing a Legendre transform on total relativistic energy. All the above tricks apply, except that rigid bodies aren't so rigid anymore.

Multivariable Calc:
- Langrage multipliers are cool.
- The Hessian determinant is useful for checking whether a stationary point is max, min, saddle, or degenerate. Take note of boundary points.
- The Jacobian determinant is useful for doing multiple integrals. It's important to set the integral up properly.
- Line integrals - find a new parameter to describe the line, and integrate along it. Remember to take the dot product.
- Surface integrals - find two parameters to describe the surface, and integrate along them. Remember to take the cross product. Be careful with the orientation, or use intuition to check the answer and reverse-engineer the sign.
- Curl of grad is zero, div of curl is zero.
- There are three different manifestations of the "fundamental theorem of calculus" -  a relationship between differentiation and integration. 1) If the vector field has zero curl, it's possible to express it as a grad of some scalar field. It's the relationship between gradient and line integral. 2) Divergence theorem expresses the relationship between divergence and volume integral. 3) Stokes' theorem expresses the relationship between curl and surface integral.

Social Psych:
- There are different ways of doing social psych research - typically analyzing data and doing experiments. There are advantages and limitations to all approaches. Note experimental biases, cultural differences, observer effects. Studies may be value-laden. They might also be conducted in unethical ways. Most studies quoted in this course are done in the US, so what is true there might not be true elsewhere.
- The "self" is a more fluid construct with East Asians than it is with Americans - i.e., different situations, different "self".
- Many biases - "Better than average" phenomenon is one of them. Tendency to warp perception/interpretation of reality to make oneself feel better.
- Situational influence -> Mechanism --> Phenomenon --> Effect. Two common mechanisms are arousal and rationalization.
- Attitudes influence behaviour, but behaviour can also influence attitudes. Some powerful persuasion techniques make use of this.
- Persuasion could take central route or peripheral route. Which route works better depends on a number of factors.
- Conformity and Obedience - [Asch, Milgram expt] Most people can be made to conform under the right circumstances (though there is always some who don't conform). A few factors that affect the degree of conformity/obedience have been found.
- Attraction - people are typically more attracted to those they meet more often and that they are more similar to.
- People form groups quite easily, and tend to bias against the outgroup.
- Stereotypes influence people's perceptions unconsciously but significantly. Approaches have been found to reduce this.

General computer science(Java)
A bit of the following:
- Elementary types and operations, Arrays, If else, loops, methods and arguments, recursion, classes and objects, scope and encapsulation, using data types, creating data types, linked lists, using interfaces, order of growth(time/resources), insertion sort vs mergesort, computability, tractability, P versus NP.

Writing seminar
- UNCLEAR THESIS/MOTIVE WILL BREAK AN ESSAY
- Read the source before critiquing it (instead of referring to someone else's critique)
- There are ways to twist a quotation to suit your purpose.
- Sometimes there's just no correct answer.
- Good stitching helps to make a essay more "flowy", which is good.

(To be continued)