David Mendoza and Kong Xiong -- An Intro to Differential Equations of Physics I like that you wanted to develop your own topic for the final project, and I applaud your ambition in choosing this one. However, in the end, I think you probably "bit off more than you could chew" here. There are a lot of points in the derivations of the wave and heat equations that are not very clear -- it's hard to see that you really understood the details or even the "big picture" of what was going on. The main weak points are: * you don't really discuss what the unknown functions in the equations mean in terms of the physics, and how they need to be considered as functions of one (or more) space coordinates, and also of time (comment 3) * The mechanics of going from the approximations to the actual PDE's are described in a rather imprecise way (comments 5,6) * There are some inappropriately sophisticated points that are touched on in ways that don't really indicate you understand what is going on (comment 2) The presentation had some of the same problems. Specific Comments: 1) pages 1 - 2: "For example, the heat equation and the wave equation both involve multiple independent variables being differentiated and related to one another." > I think it is better to say that they involve functions of several independent variables and their derivatives. You are stressing the independent variables at the expense of the quantity that is dependent on them (i.e. the displacement of the vibrating string or the temperature of the metal rod). It's those quantities that are really of interest in the physical situation. 2) page 3: The discussion of the classification of PDE into the hyperbolic, elliptic and parabolic types, plus the account of the characteristics of hyperbolic equations is rather vague here. Do you really understand what what makes a PDE elliptic or hyperbolic, what the characteristics are for the wave equation, and what the two families of curves are? This is a part of the general theory of PDE that can be treated in a more elementary way for the specific equations you are studying. 3) page 5: "In other words, d^2y /dx^2 = v^-2 * d^2 y/dt^2 (“The Wave Equation”). The independent variable v represents the point of velocity that has a constant phase while the variable y exemplifies the change of the wave passing (“The Wave Equation”)." > v is not an independent variable here. It is a constant that represents the velocity of a traveling wave. y is the independent variable that represents the dispacement of the string from its rest position. y is a function of x = position along the string and t = time. Also -- You figured out how to do the second derivatives as exponents later on. That looks better than this form! 4) page 7: I don't understand what the "Error" in the discussion of the small angle approximation means. Why are you calling this an "error"? 5) page 8: The way you get to the form of the wave equation at the top of the page here is also really unclear the way you say it here. (It was better in the presentation.) The idea is that (after using the small-angle approx.) to replace the sin(theta) terms with tan(theta) approx. = dy/dx, you are taking a difference of values of dy/dx, divided by Delta x, at two x-values separated by Delta x. Then you need to take a limit as Delta x -> 0. That's what gives the d^2y/dx^2 on the right side of the equation. 6) page 10: The "law of the mean" here is usually called the Mean Value Theorem in calculus. What it says in words is that the average rate of change of du/dx over the interval [x_0, x_0 + Delta x] is equal to some value of the second derivative d^2 u/dx^2 for some x between x_0 and x_0 + Delta x (that is what the theta' is doing here). 7) page 13: "Strang shows how to get to this solution in his video, Heat Equation, and he uses sine but it is interchangeable with cosine" > It's not really that these two functions are "interchangeable." What's going on here is that the general separable solution looks like f(t)g(x) = exp(-lambda t)(A cos(sqrt(lambda) x) + B sin(sqrt(lambda) x)) where A and B (and lambda) are arbitrary constants. The idea is that different sorts of boundary conditions can force either A or B to be zero, and then the other term does not show up. For instance, if the ends of the bar are held at temperature 0 and the bar has length L, then A = 0, sqrt(lambda) = (n pi)/L for some integer n, and only the exponential times the sin shows up in the solution. 8) page 14: "The research has a biased argument where there is absolute no counter argument that disproves how partial differential equations are embedded in physics. In addition to not presenting a counter argument, there was little evidence found on how math is important in physics." > I don't understand what you are getting at there. Why do you want to try to show the opposite of your main point, which is that mathematics is useful to describe physical situations? I think the strongest thing you could say along those lines is that some physicists might think it's possible to depend TOO MUCH on the mathematics without developing physical intuition or a true understanding of what happens in the real world situations that are being studied. In other words, they would say that the mathematics is a more or less useful tool for understanding physics but it's not the whole story. But I'm not sure you were claiming it was more than that either. What are trying to say here?? Final Project Grade Computation Bibliography: 10/10 Paper: 50/60 Presentation: 25/30 Total: 85/100