Andy Octavian's Blog

Updates on Thermodynamics TA

February 8, 2010 by Octavian

When I posted two problems (here and here) for my TA class, I found that it would be so difficult for the students to read problems and solutions via blog media. Besides the difficulty to read a large amount of equations in this blog page, it is bad idea to let them click the problems one by one. It will be a nightmare if the number of problems I write is going to increase (and it will be). So I tried to find a solution for this, and I got it.

I make a page for this TA and write all problems I solved from some books (currently from Irodov) in Thermodynamics in the pdf file, which of course can be downloaded by all of you right in that page. I have added 8 problems so far, complete with their solutions (I made it originally, so it’s still vulnerable to some calculation mistakes, conceptual misconception, typo and grammatical errors, because I often do some problem solving right before contemplating, before going to bed). I will expand the number of solved problems, and I hope you can solve those problems without seeing my solutions first. It’ll be good, because there is a possibility that you will have another approach which is different with mine, and at least, your understanding will be better if you work out those problems on your own (just make my solution as your very last reference and don’t see it except something urgent happens).

Tags: Thermodynamics
Posted in TA on Thermodynamics - Spring 2010 | Leave a Comment »

Universe without equations

February 4, 2010 by Octavian

It was yesterday that my friend Juanmarqs commented my post and pointed me about the origin of writing tensor as an indexed-object. Before that time I always think that it is clear why we write tensor as the way it is now. Since by definition tensor is the mapping from some dual vectors and some vectors (where “some” can means one or zero) to a real number, then the tensor itself is the quantity that has the “combination” of each basis dual vectors and basis vectors as its basis. Moreover, since this mapping is generally not symmetric, then if we interchange the order of those dual vectors and vectors, we will get a different number, which forces us to conclude that the combination of basis in a tensor depends on the order of dual vector spaces and vector spaces which are mapped. Using this way of thinking, the representation of tensor as the indexed-object (to be clear, the indexed-object is indeed its component, not the tensor itself) like we usually encounter is very natural.

After that my other friend recommended me about the Penrose’s diagram representation for tensors (lucky for him to have read Penrose’s The Road to Reality), in which he constructed the rule to write tensor as the electronics component-like diagrams and how to use them for tensor operation. Certainly this is remarkable, although accompanied by a little bit of pain if the number of tensors operated are full-stock. Naturally, then, I am curious about this radical idea. What if the universe is not described with equation-like system, but some other logical one which is internally consistent?

Of course we don’t have to find that because we are quite comfortable with our equation regime. But finding the best representation system of our universe in each part “locally” perhaps can help to understand it better, although it will give us a topology-like homework in the end: Why does this part use this system and that one use another? Well, just go ahead and see all possibilities.

Tags: Differential Geometry, dual vector, equation, Penrose, tensor, universe, vector
Posted in Differential Geometry, Scientist's Head | Leave a Comment »

Problem in Thermodynamics (2)

February 3, 2010 by Octavian

This is the one I pick from Irodov’s.

Two identical vessels are connected by a tube with a valve letting the gas pass from one vessel into the other if the pressure difference $\Delta p \geq 1.10$ $atm$ . Initially there was a vacuum in one vessel while the other contained the ideal gas at a temperature $t_1 = 27^\circ$ $C$ and pressure $p_1 = 1.00$ $atm$ . Then both vessels were heated to a temperature $t_2 = 107^\circ$ $C$ . Up to what value will the pressure in the first vessel (which had vacuum initially) increase?

We denote the volume of each vessel is $V$ , and that $\Delta p \geq \Delta p_0 = 1.10$ $atm$ . Since it is ideal gas, it satisfies the equation of state $pV = nRT$ . For the second vessel, initially we have the equation $p_1 V = n_1 Rt_1$ , where $n_1$ is the number of moles contained initially in the second vessel. Suppose after we heat up these two vessels to the temperature $t_2$ , we have $n_1'$ moles in the second vessel and $n_2$ in the first one, where $n_1' + n_2 = n_1$ , since this two-vessels system is closed. The equations of state for both vessels in temperature $t_2$ are

$p_1' V = n_1' R t_2$ , for the second vessel

$p_2 V = n_2 R t_2$ , for the first vessel

The difference between $p_1'$ and $p_2$ is of course $\Delta p_0$ , hence

$\Delta p_0 = p_1' - p_2 = \frac{n_1' R t_2}{V} - \frac{n_2 R t_2}{V}$

but we also have

$n_1' + n_2 = \frac{p_1 V}{R t_1}$

and by combining these two equations we will have

$p_1 \frac{t_2}{t_1} - 2 \frac{n_2 R t_2}{V} = \Delta p_0$

where we know that the second term in the LHS of the equation above is $-2p_2$ . Therefore we have

$p_2 = \frac{1}{2} \Big( p_1 \frac{t_2}{t_1} - \Delta p_0 \Big)$

which answers the problem.

Tags: pressure, temperature, Thermodynamics, vessel, volume
Posted in TA on Thermodynamics - Spring 2010, Thermodynamics | Leave a Comment »

Problem in Thermodynamics

February 3, 2010 by Octavian

Apparently this is the first problem I write for my T.A. in Thermodynamics course. Well, I just pick it from Irodov’s book. Enjoy this easy problem, before the cataclysm begins.

A vessel of volume $V = 30$ $l$ contains ideal gas at the temperature $0^\circ$ $C$ . After a portion of the gas has been let out, the pressure in the vessel decreased by $\Delta p = 0.78$ $atm$ (the temperature remaining constant). Find the mass of the released gas. The gas density under the normal conditions $\rho = 1.3$ $g/l$ .

Since it is the ideal gas, then it satisfies $pV = nRT$ . For the sake of convenience, for this problem we will write the equation of state in terms of density of gas. Since $n = m/M$ , where $m$ is the mass of that gas and $M$ is its molar mass, and by remembering that $\rho = m/V$ , we can write the equation of state as $p = \rho RT/M$ .

Then, since the temperature is constant during the process, we have $p'/\rho' = p/\rho$ , where $p'$ and $\rho'$ are the pressure and density of gas after the process, and $p$ and $\rho$ are the initial values in normal condition (STP). From this equation we have

$\rho' = \Big( \frac{\Delta p}{p} + 1 \Big) \rho$

which yields

$\frac{\Delta m}{V} = \rho' - \rho = \rho \frac{\Delta p}{p}$

and we get the answer, $\Delta m = \frac{\Delta p}{p} \rho V$ .

Tags: density, ideal gas, pressure, temperature, Thermodynamics
Posted in TA on Thermodynamics - Spring 2010, Thermodynamics | 2 Comments »

T.A. this semester: Thermodynamics!

February 2, 2010 by Octavian

I am quite happy to be the teaching assistant on Thermodynamics course for the third time in the last three years: on spring 2008, 2009 and now, 2010. Of course I am very boring with the elementary problems served in this lecture, but recently I found such a very good book that really excites me. However, I will post some elementary problems here in order to provide students the highlight problems; ones that are too good and too important to let them unwritten in this blog.

The lecturer for this course is Pak Hikam, and in his site you can download the pdf for all materials needed in this course. Sometimes he will post a homework in his site, so it’s best if you check both that site and this blog to get the information related to Thermodynamics course. Pak Hikam usually uses the book from Sears and Salinger, besides his own ppt, but of course you can use many various books available online and in the library. For the problems, I recommend books from Irodov and Lim Yung Kuo as the best compilations of problems for this course.

I really appreciate if the students (and any other readers) make a comment in my post, so I can know what’s important and know the feedback from all of you.

Posted in TA on Thermodynamics - Spring 2010 | 3 Comments »

Manifolds – Part 3

February 1, 2010 by Octavian

— 1. Flows and Lie derivatives —

In physics we are familiar with this situation: there is an area (or volume) which has the vector field which becomes the tangential vector of some curves in this area. The trivial example is a river, where the curve is the motion track of a particular dust floating down the river, and where the tangential vector is indeed the velocity of the stream. Another situation is where you draw the lines around the magnet which point from the north pole to the south pole, where the tangential vector is the magnetic field of that magnet. Due to this familiarity in physical world, it can be an advantage if we can define the similar concept in the manifold.

Suppose we have an ${n}$ -dimensional manifold with the vector field ${X}$ defined over this manifold. Provided the chart ${\{\phi, U\}}$ near the point ${p \in M}$ , then we can define the curve which has the form

$\displaystyle \frac{dx^\mu(t)}{dt} = X^\mu(x(t)) \ \ \ \ \ (1)$

where ${\{x^\mu(t)\}}$ is nothing other than ${\phi(p)}$ . We can see easily from equation above that the vector field ${X}$ is the tangential vector of this curve. We will call it the integral curve for the next discussion. There will be possibly many integral curves on the same manifold, with a given vector field. We can define the equivalence class such that two points in ${M}$ belong to the same class if they are connected to the same curve. For this, we can single out one point in each class which represents that class, to make the analysis easier for the next time.

Suppose we have one of those equivalence classes, and a point ${x_0}$ as its representation. Then, the integral curve for this point is

$\displaystyle \frac{d}{dt} \sigma^\mu(t, x_0) = X^\mu(\sigma(t, x_0)) \ \ \ \ \ (2)$

where we have changed the symbol from ${x}$ to ${\sigma}$ because we want to stress a point: that our integral curve passes the point ${x_0}$ . Then, we can make this relation

$\displaystyle \sigma(0, x_0) = x_0 \ \ \ \ \ (3)$

where we assume that the parameter ${t}$ of the curve is zero in ${x_0}$ . We also have

$\displaystyle \sigma(t, \sigma(s, x_0)) = \sigma(t+s, x_0) \ \ \ \ \ (4)$

which tells us about the freedom to choose the initial point of the integral curve, or to choose the representation for each equivalence classes we talked above. Given the vector field ${X}$ on the manifold, we can principally find the parametric equation for the integral curve by using that flow equation.

Read the rest of this entry »

Tags: Differential Geometry, directional derivative, flows, integral curves, Lie bracket, Lie derivative, tensors, vector fields
Posted in Differential Geometry, The Theory of Space and Gravitation | Leave a Comment »

How to solve physics problems

January 25, 2010 by Octavian

Due to many requests for me to write about this topic, although I am not a good problem solver, finally I try to write it down. This is a very awkward topic, because there are various problems that use different approaches, methods of thinking, and styles which depend on their creators and purposes. However, I hope we can single out the very basic principles to solve problems, the ones which can be used generally for almost any kinds of situations.

The first principle you must remember about solving the problem is understanding it. This is the very first important point, and the reason is clear. You cannot move forward to use some methods and knowledge if you even don’t understand the situation figured in that problem; if you insist, you will be wrong. This is strongly related to our creativity and our preliminary knowledge. I find many cases that the well-prepared people cannot understand the problem, because they have wrong imagination about the story told in that problem, or other cases where the high school students must solve problems about Dirac equation, which of course they cannot solve it because they even don’t know who Dirac is. This is the point: If you are stuck to understand a problem, perhaps you have missing things that are mentioned in the problem that you don’t know, or you have a wrong interpretation about it. For the first case, go away to learn more, and for the second case, just take a break and find another inspiration.

Now say that you have understood the problem completely, and it is your turn to solve it. Even after you have a clear understanding about the situation told in the problem, doesn’t mean that you can solve it easily. The very extreme example about this is the Fermat’s Last Theorem; even the children understand it clearly, but human civilization needs three centuries and a half to solve it. Of course, we don’t need to be very extreme like this. There are so many questions that are simple but require many considerations, challenges, rigorous calculations, and traps.

The methods to solve problem itself are very vast. There are many ways to see the problem, and in each perspective, we expect to have a different solution (and ideally the same answer, except for the many cases in doing the real problem, “research”). I often use 2 to 5 solutions to answer the olympiad problems, from the conceptual/qualitative consideration, rough estimate, to the most rigorous calculation (I often change the perspective and scale to generate many solutions for the same problem). Of course, it is not so easy, and I find it’s very hard to have even one solution for some hard problems (having one is very good, having two is OUCH). The sense to use the proper method and right way to go to the right answer is very hard to be developed in such a short time, so don’t even hope you can do well in your exam while you don’t practice regularly to use the methods and knowledge expertly. In this point I don’t agree with some educational institutions in Indonesia (and in other countries across the world) which use the smart solution, the fast-quick-and-efficient-way-to-remember-the-formulas method.

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Tags: how to solve physics problems, physics problems, research, research problems, solving physics problems
Posted in Scientist's Head | 2 Comments »

Multinomial Theorem, Leibniz and Taylor Formulas in Terms of Multiindex Notation

January 14, 2010 by Octavian

In this post I just wanna write an easy calculation that uses the multiindex notation for the differential operator, that was calculated when I was watching unimportant (subjectively) debates between the legislatives, government and some key persons about Century Bank’s problem. Don’t take that too seriously because I have no interest with that, and let me start our discussion today. First we introduce the multiindex

$\displaystyle \alpha = (\alpha_1, \ldots, \alpha_n) \ \ \ \ \ (1)$

where each ${\alpha_i}$ , for ${1 \leq i \leq n}$ , is a natural number. We can denote the differential operator as follows. For a function ${u: {\mathbb R}^n \rightarrow {\mathbb R} : x \mapsto u(x)}$ , then the differential of ${u}$ is

$\displaystyle D^\alpha u(x) \equiv \frac{\partial^{|\alpha|} u(x)}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}} \ \ \ \ \ (2)$

where ${x = (x_1, \ldots, x_n)}$ is the point in ${{\mathbb R}^n}$ and ${|\alpha| = \alpha_1 + \cdots + \alpha_n}$ .

— Multinomial Theorem —

Using that notation, then the theorem states

Theorem 1

$\displaystyle (x_1 + \cdots + x_n)^k = \sum_{|\alpha| = k} \binom{|\alpha|}{\alpha} x^\alpha \ \ \ \ \ (3)$

where ${\binom{|\alpha|}{\alpha} = \frac{|\alpha|!}{\alpha!}}$ , ${\alpha! = \alpha_1! \cdots \alpha_n!}$ , and ${x^\alpha = x_1^{\alpha_1} \cdots x_n^{\alpha_n}}$ , where the summation is carried over all multiindices ${\alpha = (\alpha_1, \ldots, \alpha_n)}$ with ${|\alpha| = k}$ .

Proof: We will use an induction principle. Since we know that this relation

$\displaystyle (x_1 + x_2)^k = \sum_{|\alpha| = k} \frac{|\alpha|!}{\alpha_1! \alpha_2!} x_1^{\alpha_1} x_2^{\alpha_2} \ \ \ \ \ (4)$

is true, then by assuming that

$\displaystyle (x_1 + \cdots + x_{n-1})^k = \sum_{|\alpha| = k} \frac{|\alpha|!}{\alpha_1! \cdots \alpha_{n-1}!} x_1^{\alpha_1} \cdots x_{n-1}^{\alpha_{n-1}} \ \ \ \ \ (5)$

holds for ${n \geq 4}$ , we will have

$\displaystyle \begin{array}{rcl} (x_1 + \cdots + x_{n-1} + x_n)^k &=& \sum_{|\alpha| = k} \frac{|\alpha|!}{k_0! \alpha_n!} (x_1 + \cdots + x_{n-1})^{k_0} x_n^{\alpha_n} \\ &=& \sum_{|\alpha| = k} \frac{|\alpha|!}{k_0! \alpha_n!} \sum_{|\beta| = k_0} \frac{|\beta|!}{\alpha_1! \cdots \alpha_{n-1}!} x_1^{\alpha_1} \cdots x_n^{\alpha_n} \\ &=& \sum_{|\alpha| = k} \frac{|\alpha|!}{\alpha_1! \cdots \alpha_n!} x_1^{\alpha_1} \cdots x_n^{\alpha_n} \end{array}$

and the result follows. $\Box$

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Tags: differential operator, leibniz, multiindex, multinomial, PDE, taylor
Posted in PDE | 4 Comments »

Manifolds – Part 2

January 10, 2010 by Octavian

— 1. Some objects on the manifolds —

— 1.1. Curves and functions —

When I was in the beginner physics class, I often analyzed the track of a particle on a plane that moves under some potential. Ordinarily, the curve on the plane can be parameterized using one parameter, say ${t}$ , such that every points on the curve can be marked with some definite value of this parameter ${t}$ . We also can make an arbitrary choice to place the zero of ${t}$ ; nature will not know which point you identify with the ${t = 0}$ . If I make a coordinate system on this plane (just imagine for simplicity that this is a two-dimensional plane and we use Cartesian system for the coordinate), then a point on the curve has the coordinate

$\displaystyle (x,y) = (x(t), y(t)) \ \ \ \ \ (1)$

such that if you have a single number ${t}$ , then you will have two numbers, one for ${x}$ and one for ${y}$ , and it gives you the precise position of the point on the curve (this is why we often say that curve is one-dimensional). You can make an arbitrary curve on this plane which touches every parts of this plane, and by principal, you always can identify its points using this single parameter ${t}$ .

However, this advantage to use the powerful parameter ${t}$ to mark points on the curve cannot hold peacefully if we make a generalization from the plane to a manifold. The problem is, we cannot make a global coordinate system which covers all parts of manifold. Instead, we need to define some local coordinate systems, provided that we still can transform each other using continuous mapping. Therefore, the principal that we can identify the points on the curve by using coordinate ${(x,y) = (x(t), y(t))}$ doesn’t make sense anymore, since two neighborhoods on the manifold have different convention about the values of ${x}$ and ${y}$ .

To solve this problem, we need to make two mappings. First, consider the mapping ${c: [a,b] \in {\mathbb R} \rightarrow M}$ such that ${c(s) = c(t)}$ whenever ${s = t \in {\mathbb R}}$ . We can imagine that this is the curve which connects two points on the manifold, namely ${p = c(a)}$ and ${q = c(b)}$ , and it is a simple one; there is no intersection of this curve with itself (this assumption is just for the sake of simplicity). Of course you don’t need to have finite numbers for ${a}$ and ${b}$ , the two queens of pain ( ${-\infty}$ and ${\infty}$ ) will also work. While the first mapping is the one which maps ${{\mathbb R}}$ to ${M}$ , the second mapping is ${\phi: M \rightarrow {\mathbb R}^n}$ , i.e. the mapping that serves as the coordinates marker for each points on the manifold. Therefore, we can construct the composition mapping ${\phi \circ c: [a,b] \rightarrow {\mathbb R}^n : t \mapsto \{x^1, \ldots, x^n \}}$ , such that from a single parameter ${t}$ you will get ${n}$ values of coordinates.

If the part of ${{\mathbb R}}$ is mapped to ${M}$ and produces what we intuitively call as curve, then the mapping from ${M}$ to ${{\mathbb R}}$ makes a very different object: this is simply the ordinary function defined on a manifold. Concretely, if we have a function ${f: M \rightarrow {\mathbb R}}$ , and take a point ${p \in M}$ with the coordinate ${\phi(p)}$ , then we will have the composition mapping ${f \circ \phi^{-1}: {\mathbb R}^n \rightarrow {\mathbb R}}$ . In physics, we often encounter this composition mapping ${f \circ \phi^{-1}}$ , which relates the coordinates of points on the space with a value in each points. Actually, this physicist’s view will not make any trouble if we can define a global coordinate system on the manifold, but if we just could make a system of local coordinates, then we need to distinguish the role of ${f}$ , the function from points of the manifold to some real numbers, and the role of ${\phi}$ , the attachment of points to a local coordinate system.

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Tags: coordinate system, Differential Geometry, dual space, dual vector, function, manifold, space, tensor, transformation, vector, vector space
Posted in Differential Geometry, The Theory of Space and Gravitation | 8 Comments »

Manifolds – Part 1

January 5, 2010 by Octavian

— 1. Introduction —

Let us imagine that we have a flat ordinary two-dimensional plane, which we usually use as a background, or stage, when we are figuring the motion of particles in the space. Being the background of every phenomena, whether you are aware or not, we always make the identification of each events with the point on this plane. Then, for the sake of simplicity and efficiency, we intend to make a certain coordinate system in this plane so that we can tell everyone exactly where the event happens, and when. Conventionally we use Cartesian coordinate system, the simplest system due to its orthogonality definition and because it can be taught to the 5-years-old children the way to use it. Of course, by using the strength and the efficiency of this system, we easily can identify all events on the plane, such that finally we can get the accurate and concise formulations about them.

However, nature doesn’t know what kind of coordinate system we are trying to use to figure the events on this flat plane, and how we use it. Whether we try to use the ${x}$ -axis as the horizontal one and ${y}$ -axis as the vertical, or vice versa, or the more complicated relative orientation of its axes, our universe doesn’t care about it. And so do us. We should take no care about what coordinate system we want to use to mark the events on the plane, since those events can exist without the way human figuring them. For example, if we use polar coordinate system instead, we certainly will get a different identification, or a different marking, of each events that occur on the plane, but the physics itself doesn’t change. If we observe there is a sequence nothing-collision-nothing-explosion when we are using the Cartesian, then we will observe the same sequence nothing-collision-nothing-explosion when we are using the polar. And not only this causality matter, but also other physical entities which we can observe and detect are all can be checked to be exactly the same in these two coordinate systems.

Of course, if we decide to use the Cartesian and our friend chooses the polar, there should not be any debate. Although what we get from the observation is generally different with this friend’s result, we principally can do the transformation from our Cartesian to polar and vice versa, and we conclude that our observation agrees with this friend’s result only after the coordinate transformation. This situation is similar, more or less, with the two foreigners that mention a diamond and diamants to name the same thing: the precious pure carbon. You just need an English-French dictionary to stop any possible debate between these two foreigners.

In this chapter, I will talk much about the generalization from the flat two-dimensional plane we discussed above to the more general ${n}$ -dimensional space. By constructing the theory for this general object, we should remember that we are free to choose any coordinate system, free to choose the way to use it, and the agreement only comes after the coordinate transformation.

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Tags: coordinate system, curved space, Differential Geometry, flat space, manifold, space
Posted in Differential Geometry, The Theory of Space and Gravitation | 9 Comments »

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Andy Octavian's Blog

Updates on Thermodynamics TA

Universe without equations

Problem in Thermodynamics (2)

Problem in Thermodynamics

T.A. this semester: Thermodynamics!

Manifolds – Part 3

How to solve physics problems

Multinomial Theorem, Leibniz and Taylor Formulas in Terms of Multiindex Notation

Manifolds – Part 2

Manifolds – Part 1

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