Understanding advanced mathematics.

[Randall W. Lott]

I'm not too sure how important these sort of skills are in the real world, but my professor claimed that they are used all the time by engineers dealing with electromagnetism.  He was honest and said many of us will never use it, but he's required to keep the course at a certain standard due to our department being ABET accredited.

Our University had the maximum accreditation period of 6 years, which is longer than MIT.  This means that they wont re-evaluate the EE program for 6 years due to  its sufficient "difficulty".

Either way; I want to learn as much as possible and get good at all aspects of engineering.

• I have a relatively strong understanding of integral and differential Calculus basics such as; U-substitution, integration by parts, surface area, volumes by rotation, convergence/divergence, basic series, and 2-D vectors.
  
  
• I've taken Differential Equations and have a basic understanding of the concepts.  These include Euler's Method, basic systems, variation of parameters, undetermined coefficients, basic Laplace Transforms, and basic Eigenvalues/Eigenvectors.

In September, I will be taking a course that involves Vector Calculus and Linear Algebra in one.  I've taken it before, but I had to leave the University for medical reasons and I will be taking my courses again.

All of the homework problems are usually word problems with no evidence of the methods to use, which is why it's engineering.

I just don't understand it.  How do I approach these kind of problems?  Should I create a sheet with methods and note where I should use them?

Example problem:

If R = t²a_x - t³a_y + t⁴a_z is the vector from the origin to a moving particle, find the
velocity of the particle when t = 1.  What is the component of this velocity in the direction
of the vector 8a_x - a_y + 4a_z?  What is the vector acceleration of the particle? What are the tangential and normal components of this acceleration?

The "a" terms have carrots above them, which indicated that they're unit vectors.. I think.

Don't be concerned about giving answers, because I know that some people try to have others do their coursework.  I'm trying to learn it, not cheat my way through.  I don't encourage that sort of behavior.

Thank you.

[alm]

a_x, a_y and a_z are indeed unit vectors, representing the three orthogonal axes in a three dimensional space. I'm not sure what part you have trouble with, so please ignore this if it's too basic.

The first step is to convert the question into a mathematical problem (the engineering part), and then solve this problem (the math part). To figure out the velocity at t=1, or v(t=1), first define v(t) = f(R(t)) (the velocity as function of the position). Figure out f(), then solve for R(t). Then evaluate at t=1.

This v(t=1) is a vector. The next question is about the component in the direction b = 8a_x - a_y + 4a_z. How do you define the direction of vector b? How do you determine the magnitude of vector v in the direction of vector b? Making a drawing might help. You'll be able to write the answer in terms of vector operations, and can then perform these operations, if the instructors even care about the latter part.

Once you understand the various vector operations and the problem, seeing which one applies is usually fairly straightforward in my experience. Make a list of the operations (eg. sum, difference, cross product) and draw the results with arrows if this helps your understanding. Sometimes there are multiple possible methods, like when solving differential equations. If the problem doesn't try to steer you in a particular direction, just try the simplest one that might work. Try separation of variables first, for example. If that fails (equation not separable), try a more complex method. You'll also develop some intuition about which method works best for which problem, but the beauty of the self-consistency of math is that you can use any method and end up with the correct answer.

[amspire]

To get the velocity, that is just the differential of R with respect to t.

To get the components related to a vector, that sounds like matrix calculations to me which I have been avoiding like mad ever since I left Uni.

You would need to use vector cross products and dot products of the velocity vector with the  direction vector to get the normal and tangential vectors.

Then you normalize both results to get the unit normal and tangent vectors.

Don't ask me for the exact details, as I would have to return to the textbooks myself to remember. I am very happy to leave it to you to solve. 

Richard.

Edit: I meant to say normalize the vector 8ax - ay + 4az before doing the cross and dot products, not the result.

[westfw]

>>  Vector Calculus and Linear Algebra in one.

yeah; they're sort of related.  In some ways, this is more important than the details of how to solve particular forms of integrals/etc, because this is the way that nice, pure, math is mapped onto 3d reality.  I had a blissful year or so at college when math, physics, and engineering, and maybe even chemistry) classes were all doing this at the same time.  I wish I had appreciated it at the time!

That said, the last time I looked at my college texts, I pretty much couldn't even recognize the symbols anymore.

In your problem, you have an equation for the position of a particle, and you should know from freshman physics that the velocity is the first derivative of position.  Except now you have an equation in 3D, so it becomes a little more complicated.  You might have done this earlier by dividing the equations into pieces for each coordinate.  And in fact you'll end up doing that, but you'll have cleaner notation.  In a large sense, the vectors and linear algebra are ways have having more general concepts to apply to a situation, like the way you learn that velocity is the derivative of position rather than memorizing special cases (s=1/2 at^2, v=at, etc)  So once you get comfortable with it, it should be easier to apply than your current math...

And yes, it comes up all the time in E&M.  Gauss's law: (Del dot E = rho / epsilon0) would be a much more obnoxious looking equation without the vectors...

The 2nd part I don't remember how to do at all, but I assume there's a standard equation...

[Randall W. Lott]

Thank you for the responses.

The first part isn't too bad actually.  It just looks scary!

[IanB]

I just don't understand it.  How do I approach these kind of problems?  Should I create a sheet with methods and note where I should use them?

There are many different ways to approach this question, so I'll try a few of them.

Firstly, there isn't really such a category as "advanced mathematics" as opposed to "not advanced mathematics". Mathematics is like a kind of staircase, with each step building on top of the steps that went before and adding a few new things each step. So if you already at a given step then the next step doesn't seem too hard to climb, but if you are looking several steps up at once it seems like a dizzy height.

Therefore one thing to do when a piece of mathematics seems complicated and confusing is to identify the steps that came before, then go back and review those first to make sure they are clear. To use a kind of analogy, in the US high school system there is "pre-calculus" that is a necessary precursor to "calculus". It is intended that you have all the pre-calculus stuff down pat before you try to learn calculus, otherwise calculus will be too difficult. You can argue the merits of this particular example, but calculus is going to be impossible if you can't do ordinary algebra.

A second response to the question is to observe how mathematics is really all about recognizing patterns, making connections between the patterns, and seeing where the patterns can be applied. This is really hard to do without practice. So doing lots of practice problems on the earlier steps is important to really get them solidified in your mind.

On the specifics of your example problem, one pattern is that "velocity is the first derivative of position wrt time". So if you have an equation that gives the position of a particle as a function of time (you have to pick up that t is time in this example), then differentiating that function will give the velocity as a function of time. A second pattern is that differentiating vector functions wrt time can be done by differentiating each component of the vector individually. A third pattern is that "the component of a vector in the direction of another vector" is a standard pattern in linear algebra. As soon as you see that question you have to say to yourself, "Aha! I know there is a standard way to find that." Then if you can't remember it, go off to your linear algebra textbook and look it up.

It continues with the other parts of the question. Like acceleration being the second derivative of position, or the first derivative of velocity, wrt time. Calling it a "vector acceleration" is kind of redundant; acceleration will be a vector just like the velocity was. "Tangential and normal components" again is a standard pattern. You either know the formula or you don't, but if you don't you can go look it up in the textbook once more.

I hope this helps a bit. Things become less scary as familiarity grows. If you look at something and it seems scary, it tells you that you need more review and more practice. If it seems scary and you have an exam tomorrow you are in trouble. If it seems scary but you are only just beginning the course, then expect enlightenment to come as you work through the problems and practice each concept.

For my part, I couldn't answer that question satisfactorily if I were sitting in an exam today. I don't remember all the necessary formulas. But as I read the question, I feel confident I could answer it if I went and reviewed the material. That's really the thing; recognizing the difference between not having a screwdriver in your toolbox (but you know where to get one), or not even knowing that you need a screwdriver. Once you know you are going to need a screwdriver the problem is half solved.

[Randall W. Lott]

Thank you.  Unfortunately, we didn't have a text book for the course.  Most students love hearing that because you can save a few hundred dollars, but it would have been helpful.

I now understand the basic concept of taking the derivative to find the velocity in the x, y, or z direction.  It's not as bad once I understood that the function was expressing a position.

I mentioned that I have a decent understanding of fundamental calculus.  Knowing when I can use it is the hard part 

I'm getting prepared early.  Doesn't hurt, eh?

Are there any tips for Fourier Series?  Is that considered Linear Algebra, Vector Calculus..?  I studied it in a "Signals and Systems" course.  That was super confusing since I've never heard of a LTI until then.

I would like to be in a position to solve or understand anything that comes my way..  eventually.

[Bored@Work]

Knowing when to use what math comes from understanding the problem, chopping it into smaller subproblems, deciding how to solve each subproblem divide et impera, and stringing the subproblem solutions together. This in turn requires to know the properties of the math operations.

[westfw]

>> Are there any tips for Fourier Series?
Signals and Systems sounds about right.  It's pretty specialized to EE and signal processing.
One of the things that happens in EE is that the tricky parts of the math end up "going away", because everything is (treated as) a complex exponential or sinusoid, which have "simple" derivatives and integrals.  You have to take the two years of math so that you can understand why you can throw it away and use Phasors and Smith Charts instead... (And Fourier Series are just a way to convince yourself that even things that don't look like sinusoids can still be treated that way.)  (don't listen to me.  I didn't like E&M much, and went on to become a, a, ... *Programmer*.  Shudder.)

[nctnico]

I have the same situation. Every now and then I get to solve a math problem. The problem is that a lot of textbooks deal with the dry theory and no real world applications. However, many years ago I bought the books 'modern engineering mathematics' and 'advanced modern engineering mathematics'. I'm quite sure these books cover every math problem you'll ever encounter during your EE carreer.