# I want to learn relativity and 'rocket science'. What courses do I need to take? **URL:** \ **Category:** Factual Questions\ **Created:** [November 6, 2009, 5:11pm UTC](https://boards.straightdope.com/t/i-want-to-learn-relativity-and-rocket-science-what-courses-do-i-need-to-take/516600 "2009-11-06T17:11:20Z")\ **Posts on this page:** 7\ **Page:** 2
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 7, 2009, 1:33am UTC](https://boards.straightdope.com/t/i-want-to-learn-relativity-and-rocket-science-what-courses-do-i-need-to-take/516600/21 "2009-11-07T01:33:04Z")
With the caveat that most of those elements probably aren’t independent. The Riemann tensor, for instance (which is the only fourth-rank tensor you’re likely to encounter) has so many relationships between its various elements that it only actually has 20 independent elements. ---
**Author:** ![JWT\_Kottekoe](https://avatars.discourse-cdn.com/v4/letter/j/3ec8ea/32.png) [@JWT\_Kottekoe](https://boards.straightdope.com/u/JWT_Kottekoe)\ **Post date:** [November 7, 2009, 2:59am UTC](https://boards.straightdope.com/t/i-want-to-learn-relativity-and-rocket-science-what-courses-do-i-need-to-take/516600/22 "2009-11-07T02:59:09Z")
> [@Nancarrow](#): > > I don’t think you can get General Relativity without tensor. There is a way to do it without tensors. You can use the “exterior calculus” using so-called “differential forms”. It is a beautiful way to do the mathematics, but harder to understand than tensors. > [@Chronos](#): > > With the caveat that most of those elements probably aren’t independent. The beauty of differential forms is that all of the symmetry of the tensors is built in to the differential forms themselves. Everything is totally antisymmetric, so you don’t have a bunch of redundant matrix elements as you do in the tensor formulation. You also discover why the cross product of two vectors is not really a vector, it just has the same number of independent components as a vector. ---
**Author:** ![Sunspace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sunspace/32/1250_2.png) [@Sunspace](https://boards.straightdope.com/u/Sunspace)\ **Post date:** [November 7, 2009, 3:25am UTC](https://boards.straightdope.com/t/i-want-to-learn-relativity-and-rocket-science-what-courses-do-i-need-to-take/516600/23 "2009-11-07T03:25:21Z")
> [@DrCube](#): > > I’m wondering how he took DiffEq and didn’t learn any matrix algebra. Does “Cramer’s Rule” ring a bell? Actually, we did nibble around the edges of matrices, now that I think about it. And we did Laplace transforms in third year (time domain to frequency domain), but they were basically black magic. I was taking electronics engineering technology, a three-year diploma at community college rather than a four-year engineering degree at university. So while we touched on a lot of stuff, we never got as deeply into the theory as engineering students at university did. Here’s the [present course at the same institution](http://www.sheridanc.on.ca/Programs%20and%20Courses/Full-Time%20Programs/Programs%20A-Z%20Index/Electronics%20Engineering%20Technician%20-%20Technology.aspx), but of course it’s changed in the 24 years since I graduated. And Sheridan has as well… it now offers some university courses, calls itself an ‘Institute of technology and advanced learning’, and I expect it will morph into a university in the next ten years or so. Basically following the same road that Ryerson University in Toronto did. ---
**Author:** ![Stranger\_On\_A\_Train](https://avatars.discourse-cdn.com/v4/letter/s/13edae/32.png) [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)\ **Post date:** [November 7, 2009, 5:28pm UTC](https://boards.straightdope.com/t/i-want-to-learn-relativity-and-rocket-science-what-courses-do-i-need-to-take/516600/24 "2009-11-07T17:28:40Z")
> [@thirdname](#): > > Aren’t space missions usually calculated with Newtonian physics? Yes. Relativistic corrections are needed for high precision satellites, i.e. GPS, GLONASS, and Galileo navigation satellites, but for space probes and most communication and surveillance satellites any relativistic influences are masked by other measurement errors and gravitational perturbations, and are addressed by en route trajectory corrections. Although spacecraft navigation and mission planning are really, really difficult, multi-valued “wicked” problems, plotting an intercept trajectory or a transfer to a defined orbit is really pretty easy, and plotting a two-body conic section orbit requires only planar geometry (as long as you define your coordinate system conveniently). Prussing’s _[Orbital Mechanics](http://www.amazon.com/Orbital-Mechanics-John-E-Prussing/dp/0195078349)_ is the best general reference I’ve found for this which should be accessible on the undergraduate level. Thomson’s _[/url=[Introduction to Space Dynamics (Dover Books on Aeronautical Engineering): William Tyrrell Thomson: 8580001359137: Amazon.com: Books](http://www.amazon.com/Introduction-Dynamics-William-Tyrrell-Thomson/dp/0486651134/ref=pd_sim_b_2)]Introduction to Space Dynamics_ is pretty good and readily accessible reference. “Rocket science” is more than just orbits, of course; the details of propulsion systems are critical to plotting the trajectories of real spacecraft, and making space vehicles with mechanisms than can deploy instruments and solar panels, and function reliably for years without any ability for repair or maintenance requires a lot of effort and testing. There are several different technical disciplines in and of themselves, and each of them would require years of schooling and experience to have even a basic functional grasp. Stranger ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [November 7, 2009, 6:39pm UTC](https://boards.straightdope.com/t/i-want-to-learn-relativity-and-rocket-science-what-courses-do-i-need-to-take/516600/25 "2009-11-07T18:39:39Z")
> [@Nancarrow](#): > > 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. It doesn’t look like anyone’s given the full answer to what tensor is. To understand what a tensor is, you only need to understand what (1) a vector space is, and (2) what a linear map is. I’ll assume that you know what a [vector space](http://en.wikipedia.org/wiki/Vector_space) is. A linear map is then just a map between vector spaces that respects the vector operations, which are vector addition and scalar multiplication. That is, a map _L_ between vector spaces _V_ and _W_ is a map such that _L_(_u_ + _xv_) = _L_(_u_) + _x_ _L_(_v_) for all vectors _u_ and _v_ in _V_ and scalars _x_. (You often hear that linear maps _are_ matrices, but this is misleading. _If_ you fix bases for all your vector spaces, _then_ every linear map corresponds to a specific matrix. But the linear map and the matrix aren’t the same thing. It’s better to think of the matrix as a computational device that lets you see how the linear map acts. By analogy, consider the functions that compute the positive integer powers of real numbers. For each positive integer _n_, there is a function _f_[sub]_n_[/sub] such that _f_[sub]_n_/sub = _x_[sup]_n_[/sup]. But the function _f_[sub]2[/sub] isn’t _equal to_ the number 2; it just _corresponds to_ the number 2.) Once you know what a linear map is, the concept of a _multilinear_ map is easy. Let _V_[sub]1[/sub], . . . , _V_[sub]_n_[/sub] be some finite sequence of vector spaces. (This sequence can contain multiple copies of one vector space.) Call these the _source_ spaces. Let _W_ be another vector space, called the _target_ space. A multilinear map _M_ with these sources and this target is a function that takes in a vector _v_[sub]1[/sub] from _V_[sub]1[/sub], _v_[sub]2[/sub] from _V_[sub]2[/sub], . . . , and _v_[sub]_n_[/sub] from _V_[sub]_n_[/sub], and then produces a vector _M_(_v_[sub]1[/sub], . . . , _v_[sub]n[/sub]) = _w_ in the target space _W_. Moreover, (and this is what makes _M_ _multilinear_) if you fix all but one of the vectors from the source spaces, _M_ acts linearly on the remaining argument. That is, if you fix _v_[sub]1[/sub], . . . , _v_[sub]_i_ − 1[/sub], _v_[sub]_i_ + 1[/sub], . . . , _v_[sub]_n_[/sub] then the vector _M_(_v_[sub]1[/sub], . . . , _v_[sub]_i_[/sub], . . . , _v_[sub]_n_[/sub]) in _W_ is a linear function of _v_[sub]_i_[/sub]. The final new concept you need is that of the _dual space_ of a vector space. If _V_ is a vector space, then the dual space _V_\* is just the vector space of linear functions from _V_ to _F_, where _F_ is your field of scalars (usually the real or complex numbers). This is all we need to define a _tensor_. Fix a vector space _V_. Then a tensor _T_ is a multilinear map whose target space is the field of scalars, and whose source spaces consist of several copies of the dual space _V_\* followed by several copies of _V_ itself. That is, the sequence of source spaces looks like _V_\*, . . . , _V_\*, _V_, . . . , _V_, and the target space is _F_, the field of scalars. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 7, 2009, 7:47pm UTC](https://boards.straightdope.com/t/i-want-to-learn-relativity-and-rocket-science-what-courses-do-i-need-to-take/516600/26 "2009-11-07T19:47:40Z")
> [@Tyrrell\_McAllister](#): > > This is all we need to define a _tensor_. Fix a vector space _V_. Then a tensor _T_ is a multilinear map whose target space is the field of scalars, and whose source spaces consist of several copies of the dual space _V_\* followed by several copies of _V_ itself. That is, the sequence of source spaces looks like _V_\*, . . . , _V_\*, _V_, . . . , _V_, and the target space is _F_, the field of scalars. The mathematician in me wants to say that this final definition is only correct when V is finite-dimensional (so that we can identify the tensor product (and V itself) with its double dual). But this thread isn’t really for the mathematician in me. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 7, 2009, 8:01pm UTC](https://boards.straightdope.com/t/i-want-to-learn-relativity-and-rocket-science-what-courses-do-i-need-to-take/516600/27 "2009-11-07T20:01:28Z")
Still, I think it’s worth noting that the interesting thing about tensors is that they themselves form a vector space. So another way to put it is that, for any bunch of vector spaces A, B, C, etc. [as few or as many as you like], there is some essentially unique vector space T [the “tensor product” of A, B, C, etc.], with the property that linear maps out of T naturally correspond to multilinear maps out of A, B, C, etc. That is, for any multilinear map from A, B, C, etc., into some space Y, there is a corresponding linear map from T into Y, and vice versa, and furthermore this correspondence is “natural” in the sense that postcomposing the same linear map to corresponding maps yields corresponding maps. So with the aid of the tensor product space, instead of having to worry about multilinear functions as such, we can readily treat everything as just a linear function on the right domain. And, in general, when scientists speak of a “tensor”, they mean an element of the tensor product of a bunch of copies of some particular vector space and of its dual space. [Previous page](https://boards.straightdope.com/t/i-want-to-learn-relativity-and-rocket-science-what-courses-do-i-need-to-take/516600.md?page=1)