First off, sorry this is past the midnight deadline. Monday and Tuesday are absurdly busy for me on my schedule. I've forgone proper symbols to get this out quicker; my apologies for the messiness.
Section 1.6: Cool stuff - derivatives are straightforward, the easy proofs with Laplacian, curl, and divergence are impressive. Gets a little unclear towards the end. (1.89) is not manifestly covariant since it is not yet expressed in terms of tensors and we do not know how it changes under rotation. Is covariant here used in the sense that it transforms nicely under rotations or in the sense of covariant tensors/pseudovectors. But is B_ij a tensor or a pseudotensor? It is not obtained by tensor product or by Levi-Civita contraction, so we have not yet established a rule for determination.
Section 2.1-2.2: One question that immediately stands out is equation (2.1). We have a volumetric force F_i multiplied by mass (rho*dV) equal (in units) to the net force. That is, the units seem not to work out. How is this resolved? Another point lies in equation (2.6) - why are we computing surface integrals when we are told to integrate over volume? The rest pretty much made sense, though I'll be interested to see how this is presented in class. My mathematical instincts are a little frustrated by the "wishy-washiness" of the argument.
Section 2.3: Okay so F_i, though not mentioned before, is force per unit mass. That makes more sense now. This stuff is very cool and intuitive. Everything else pretty much makes sense.
Good stuff.
Tuesday, January 29, 2008
Sunday, January 27, 2008
Reading 1/28/2008: 1.1-1.5
Sections 1.1,1.2: Okay, so these are basically what we did in class. Fairly simple stuff to those who have seen tensors. The mathematician in me is still freaked out by the index stuff, and the physicist is wondering what happened to superscript/subscript Einstein notation used in GR. But otherwise, it's pretty intuitive. That said, it does seem the notation is incredibly cumbersome in certain cases. For example, equation 1.31
||aij||2 = ||aij|| ||aij|| = ||aij ajk|| = ||&deltaij|| = 1
seems vastly more clear when written as
|A|2 = |A| |A| = |A| |A|T = |A| |A|-1 = 1
As the middle step in index notation does not seem easily justified, whereas we know that orthonormal matrices have transpose inverses. It's a minor point, though.
Section 1.3: The subscript issue was nicely resolved here, though I wish there were a more rigorous definition of contravariant vs. covariant. I'll need to go dig that up on Wikipedia. It seems to be an issue that doesn't come up as prominently (if at all) in the mathematical notation. Contravariant vectors transform 'like' coordinates and covariant transform 'like' the gradient, but what exactly does that mean? I realize this won't be necessary for the class, but it's intriguing. Another point (definition-wise) I'd be curious about here is exactly what it means to be an isotropic tensor - I'm presuming this means that its an eigenvector of the rotation matrix with eigenvalue one.
Section 1.4: Pseudovectors are intriguing. I haven't seen them in a mathematical context and am wondering if they have a common analogue, perhaps under a different name. Mathematicians, of course, aren't looking at things from the same perspective - cross products are not as prominent and the central focus isn't how tensors transform under transformations. That said, what is the fundamental difference? And where does the name tensor density come from - in particular, why density? I expected this to be something like a tensor field, but obviously it is not.
Section 1.5: This is pretty much the standard tensor stuff. It's actually a little strange to see contraction without the need to raise or lower indices, but it makes sense given the Cartesian focus. The transition between tensors and tensor densities is definitely new, however, though I'd still like to know what is fundamentally going on - at least in the mathematical realm - with densities. Somehow, that usually makes the physics make more sense.
So we're still doing fundamentals. Tensors are cool little creatures, and it should be good to see how they work in this context, as opposed to GR or Diff. Geo.
||aij||2 = ||aij|| ||aij|| = ||aij ajk|| = ||&deltaij|| = 1
seems vastly more clear when written as
|A|2 = |A| |A| = |A| |A|T = |A| |A|-1 = 1
As the middle step in index notation does not seem easily justified, whereas we know that orthonormal matrices have transpose inverses. It's a minor point, though.
Section 1.3: The subscript issue was nicely resolved here, though I wish there were a more rigorous definition of contravariant vs. covariant. I'll need to go dig that up on Wikipedia. It seems to be an issue that doesn't come up as prominently (if at all) in the mathematical notation. Contravariant vectors transform 'like' coordinates and covariant transform 'like' the gradient, but what exactly does that mean? I realize this won't be necessary for the class, but it's intriguing. Another point (definition-wise) I'd be curious about here is exactly what it means to be an isotropic tensor - I'm presuming this means that its an eigenvector of the rotation matrix with eigenvalue one.
Section 1.4: Pseudovectors are intriguing. I haven't seen them in a mathematical context and am wondering if they have a common analogue, perhaps under a different name. Mathematicians, of course, aren't looking at things from the same perspective - cross products are not as prominent and the central focus isn't how tensors transform under transformations. That said, what is the fundamental difference? And where does the name tensor density come from - in particular, why density? I expected this to be something like a tensor field, but obviously it is not.
Section 1.5: This is pretty much the standard tensor stuff. It's actually a little strange to see contraction without the need to raise or lower indices, but it makes sense given the Cartesian focus. The transition between tensors and tensor densities is definitely new, however, though I'd still like to know what is fundamentally going on - at least in the mathematical realm - with densities. Somehow, that usually makes the physics make more sense.
So we're still doing fundamentals. Tensors are cool little creatures, and it should be good to see how they work in this context, as opposed to GR or Diff. Geo.
Tuesday, January 22, 2008
First Post
Well, here I am posting on yet another blog. Hello world! f1r2t p0st! Digg me! And with that out of my system, I move on.
So in case it wasn't extraordinarily obvious from my oh-so-creative blog title, I'm Ben Preskill. And I'm a junior at Harvey Mudd College with majors in Math and Physics and a concentration in Economics.
Truth is, I like math. All of it. And I especially like applications of math to cool problems. But I have no idea which applications excite me the most. My solution? Try as many as I can. Fluid mechanics has to be one of the most prominent areas of applied math out there. It's used as a paradigm for applications of calculus, differential geometry, and PDEs, and so I figured I damn well ought to see what this stuff is all about. That's pretty much why I'm taking Continuum and Fluids. Oh, and tensors rock.
Really, I don't know what to expect. I know there are tensors involved and I know we'll be modeling...well, fluids. But that's sort of the point - I don't know anything about fluid mechanics, really, and I want to know what the subject is all about. Moreover, I've seen a lot of math from a physics perspective (being a physics major,) but haven't seen as much physics from a math perspective. Although I don't expect the whole course to be the latter, I'm guessing I'll get a taste of it. Finally, I imagine continuum mechanics makes modeling almost any type of macroscopic physical situation a lot more feasible. That's just a guess, though. For all I know, you might mostly end up with some god-awful system of PDEs that requires ode15s because it's too damn stiff, and then MATLAB takes an hour to spit out a vector containing a bunch of numbers to the negative fiftieth power. But I hope not.
Now, I've been asked about a favorite equation, and I've not put a lot of thought into this. The physicists would love me if I said Schroedinger or a Hamiltonian System. But then, I'm so tempted to pick something powerful and abstract from math, like some kind of one-line statement of the Riesz Representation Theorem. I could also be cute and talk about how sexy Navier-Stokes is. Instead, how about something that underlies it all. If f ∈ C1 where f : ℜ → ℜm , we can write
f(t) = ∑ ck eikt
Now that's sexy. But Fourier series are pretty passe, I'll admit, so I'll also throw in the geodesic equation
d2xk/dt2 + ∑ij Γkij dxi/dt dxj/dt = 0
Wow, not exactly super pretty. I guess I'll just do that LaTeX thing in the future.
Anyway, that's just about it. For a fun fact about myself I'll throw in that I'm learning guitar now. For even more exciting detail, I'll confess that my inspiration is an unyielding desire to play early 90's britpop such as, e.g., Oasis. No, I'm not kidding.
Expect more fun to follow. Cheers.
So in case it wasn't extraordinarily obvious from my oh-so-creative blog title, I'm Ben Preskill. And I'm a junior at Harvey Mudd College with majors in Math and Physics and a concentration in Economics.
Truth is, I like math. All of it. And I especially like applications of math to cool problems. But I have no idea which applications excite me the most. My solution? Try as many as I can. Fluid mechanics has to be one of the most prominent areas of applied math out there. It's used as a paradigm for applications of calculus, differential geometry, and PDEs, and so I figured I damn well ought to see what this stuff is all about. That's pretty much why I'm taking Continuum and Fluids. Oh, and tensors rock.
Really, I don't know what to expect. I know there are tensors involved and I know we'll be modeling...well, fluids. But that's sort of the point - I don't know anything about fluid mechanics, really, and I want to know what the subject is all about. Moreover, I've seen a lot of math from a physics perspective (being a physics major,) but haven't seen as much physics from a math perspective. Although I don't expect the whole course to be the latter, I'm guessing I'll get a taste of it. Finally, I imagine continuum mechanics makes modeling almost any type of macroscopic physical situation a lot more feasible. That's just a guess, though. For all I know, you might mostly end up with some god-awful system of PDEs that requires ode15s because it's too damn stiff, and then MATLAB takes an hour to spit out a vector containing a bunch of numbers to the negative fiftieth power. But I hope not.
Now, I've been asked about a favorite equation, and I've not put a lot of thought into this. The physicists would love me if I said Schroedinger or a Hamiltonian System. But then, I'm so tempted to pick something powerful and abstract from math, like some kind of one-line statement of the Riesz Representation Theorem. I could also be cute and talk about how sexy Navier-Stokes is. Instead, how about something that underlies it all. If f ∈ C1 where f : ℜ → ℜm , we can write
f(t) = ∑ ck eikt
Now that's sexy. But Fourier series are pretty passe, I'll admit, so I'll also throw in the geodesic equation
d2xk/dt2 + ∑ij Γkij dxi/dt dxj/dt = 0
Wow, not exactly super pretty. I guess I'll just do that LaTeX thing in the future.
Anyway, that's just about it. For a fun fact about myself I'll throw in that I'm learning guitar now. For even more exciting detail, I'll confess that my inspiration is an unyielding desire to play early 90's britpop such as, e.g., Oasis. No, I'm not kidding.
Expect more fun to follow. Cheers.
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