Over in this thread, they’re dioscussing relativity.
Here I stand in my sock feet. I have a lay knowledge of things like physics and astronomy, but none of the math (I’ve taken math up to basic differential equations). I’d like to plot my own orbits across the solar system, and calculate how space bends, and so on.
What courses do people take to understand this? People netion ‘tensors’ and ‘matrix algebra’. I’m wondering whether there’s a well-known sequence of things toi study, like ‘regular high-school math > differential calculus > integral calculus > ?? > ?? > relativity’
Well, I don’t know anything about rocket science, but special relativity requires nothing more than basic algebra and a little calculus, and is usually taught in first year college physics courses, or AP physics in high school.
Basic physics also covers how to solve (Newtonian) two-body problems, which is sufficient for plotting a large variety of space missions.
General relativity is much harder, and involves tensors and frobnitzes and all that other stuff that’s way beyond most mortals.
If it’s any consolation* I did physics at university (a rather good one too), I have a heavy tome on Maths for Physics with a whole chapter called “Tensors”, not to mention Feynman’s lectures on physics, which contain a chapter called “Tensors”, and in the seven years since graduating, thinking about the subject, I still have no idea what a tensor IS, other than an array of numbers with some weird rules for what you do with them. I gather there are schools of mathematical thought that say that’s quite enough to say what tensors are. Bah.
Unfortunately I don’t think you can get General Relativity without tensors, but Special Relativity you can do with just algebra (and a little calculus if you want to be fancy)
When people mention “matrix algebra”, they’re referring to “linear algebra”, which is useful for solving systems of equations , both algebraic and differential. You will probably want to look into classes/textbooks covering (aero)dynamics, thermodynamics, fluid mechanics, and then specific material on astrodynamics and rocket propulsion.
If you’ve already had DiffEQ, then you’re doing pretty well on the math background you’ll need. A course on matrix algebra would be useful, but not essential. As for the tensors themselves, the hardest part about them is realizing how easy they are.
For the love of Og, though, make sure that your first exposure to tensors is not from an engineering class. Engineers make tensors much, much harder than they need to be.
Quoth Nancarrow:
If it’s any consolation, by that standard you don’t know what vectors are, either. You’ve just become familiar enough with them that you don’t mind the fact that you don’t know what they are. So it is with tensors as well. Ultimately, everything comes down to scalars, and vectors and tensors are both just tools you use to get a scalar answer in the end.
I always imagined a tensor as a way to describe a field in three dimensions. A field would have three Cartesian coordinates at any one geometric point in three dimensions. So a tensor describing an entire field must have a three coordinates for space and three coordinates for the size and direction of the field.
Am I right? It sounds pretty simple, but I’m not sure I would know how to deal with them mathematically. I would love to learn, since non-linear optics seems to use them heavily, but I made the mistake of never taking linear algebra.
A simple, easily-grasped, example of a tensor might be the stress-strain tensor. Stress is basically a force (actually, a force over an area), and strain is a deformation of something caused by the stress. So the stress-strain tensor for a particular material describes how the material responds to forces on it.
For a simple material, the response will always just be in the same direction. If I push on the left and right sides of a block, the block gets a little narrower in the left-right direction. If it were always so simple, I could just use a scalar for the stress-strain relation. But I might also have a material where, if I squeeze it in the x direction, it not only gets a little smaller in the x direction, but also bulges out in the y or z directions. I might even have something that bulges in both, but bulges more in the z direction than in the y, and so on. To describe all of these complicated distortions resulting from the force, I need something more complicated than a scalar. That’s what the stress-strain tensor is.
My kid’s an AE major.
I believe last year he had a Physics course that was largely SR.
He said the intro to AE course was lots of fun - not nearly as much heavy lifting as the rest of the reqs.
I took Diff EQ, Vector Calc, Linear Algebra and Discrete Math, and tensors were never covered. (I did have a tutor who told me that tensors were multi-dimensional vectors, but that is about it). That was all twenty years ago and the most advanced math I have used outside of college was basic trig. I once tried to understand how qubits worked, but I had no idea what they were talking about. (Hilbert spaces, Lorentz invariance and bra-ket notation were not covered in school).
As far as the OP goes, doesn’t rocket science cover more than orbital mechanics?
You did actually do tensors, you just didn’t know that’s what you were doing. A matrix is a kind of tensor. But tensors per se aren’t usually covered until you get to a course where they’ll be used, like relativity.
The number of dimensions is a completely different question than the rank. A vector, with any number of dimensions, is a first-rank tensor. A matrix is a kind of second-rank tensor (strictly speaking, it’s actually a representation of a tensor, but close enough for our purposes). You could have a 2x2 matrix, which would be a two-dimensional second-rank tensor, or you could have a 10x10 matrix, which would be a ten-dimensional second-rank tensor, or you could have a vector with 7 elements, which would be a five-dimensional first-rank tensor.
I’m wondering how he took DiffEq and didn’t learn any matrix algebra. Does “Cramer’s Rule” ring a bell? I don’t remember what it is, but maybe you heard about the Wronskian?
When I took differential equations, most of the second half of the semester was about solving systems of equations, using matrices and linear algebra. It wasn’t my first exposure to linear algebra, but it was for at least a few people in the class, because the professor always grumbled about how linear algebra should have been a pre-req.
I remember liking that class a lot, but the only way I can solve differential equations now is with Laplace transforms.
Also, I used to have a free PDF from the internet that I tried to teach myself tensors from. It was really clear and I was beginning to think they were easy to understand when life got all busy and it fell by the wayside. I’ll see if I can dig it up.
Right. If you want to launch rockets and space probes through the solar system, Newtonian gravity will work perfectly fine. Things might go a little awry if you go too close to the sun (for example, Mercury’s orbit does something weird that Newton’s laws can’t explain), but every space mission thus far just used Newtonian gravity.
It’s only if you want to put something in orbit around a neutron star or a black hole that you really need general relativity, because orbits work differently there, but as far as I know there aren’t any of those things in the solar system, so for now it’s all just hypothetical. The only actual application (outside of observational astronomy) that I know of for spacetime curvature calculations is keeping GPS accurate.
With tensors, how you determine the rank versus number of dimensions is that the rank is determined by how many numbers you need in order to refer to a specific element of the tensor.
For example, R[sub]ab[/sub] is an element of a rank 2 tensor, where a is what row you’re at, and b is what column you’re at. The number of dimensions is simply how many rows or columns you have.
Meanwhile, R[sub]abcd[/sub] is an element of a rank 4 tensor, because you need to give values for a, b, c, and d in order to specify which element of the tensor you’re referring to. In general relativity, tensors are typically 4 dimensions, 3 for space and one for time, so that tensor would have 4[sup]4[/sup]=256 separate elements.
