2
$\begingroup$

Next semester I have the possibility to attend a course with the above title. But I'm not sure if I should, because I looked on Wikipedia and Princetions Companion to Mathematics and couldn't find anything the tells me what this subject is really about, what problems it solves etc. and a course description presently is nonexistant. Generally searching on the web only returned links to monographs which venture directly into details and don't describe the "broad picture".

So I thought I ask you!

Complementing the above questions about the big picture into which this topic fits I'm specifically interested

  • if it would have been wise to have heard first a "Calculus of Variations" course, to pick up the main ideas, since the "in $L^p$ spaces" appendix indicates that this is a specialized subbranch of it;

  • what kind of tools this branch of mathematics provides and in which other branches I can use them ? (It may be of use knowing that I won't specializing be in more applied branches of maths like optimization or numerical maths...)

Last: If someone has a reference answering all these question, I'd be happy to consult that, instead of making someone type a complete answer (though I certainly won't mind that either).

$\endgroup$
2
  • $\begingroup$ Perhap's you could politely ask the instructor? $\endgroup$ Commented Jan 29, 2014 at 16:50
  • $\begingroup$ @DavidMitra For different reasons, this option is at the present time (when I have to make the decision of whether to take the course or not) not available to me. $\endgroup$ Commented Jan 29, 2014 at 16:51

1 Answer 1

Reset to default
4
$\begingroup$

Broadly, this course will develop and use a collection of theoretical tools used to minimize or maximize a given functional which is defined on some function space (which is usually $L^p$ -- more on that below). If we let $V$ be some function space, then you'll look at finding the element $v \in V$ such that a functional $L: V \to \mathbb{R}$ attains its local (or global) minimum or maximum value at $v$. If you think of using calculus to minimize or maximize a function $f:\mathbb{R}^n \to \mathbb{R}$, calculus of variations (roughly) examines the infinite-dimensional version of such problems. Techniques from calculus of variations are directly used in physics and optimal control theory very frequently.

It's hard to answer the other questions for sure without knowing the course structure of your university. However, to me the phrase "Calculus of Variations in $L^p$ Spaces" might as well say "Calculus of Variations." As I mentioned above, the goal of this course is to find extrema of a given functional defined on a function space. Very commonly (I'm tempted to say "almost always") the function space of interest is some $L^p$ space. If your university offers this as a second course in a two-course sequence on calculus of variations, then naturally I'd take the first one. If not, then I think taking this one is fine since appending the "in $L^p$ spaces" doesn't appear to substantially change what one might expect from a course simply titled "Calculus of Variations."

In this class you'll use $L^p$ spaces quite a bit (though I suppose that's obvious from the course name) and you'll likely work quite a bit with $L^2$ in particular, so you'll learn some amount about Hilbert spaces in addition to other function spaces. You'll also likely get introduced to some functional analysis as well. The depth to which you'll explore each of these topics (or potentially some others) depends heavily on the instructor.

Edit: Just to give you a small glimpse at what you'll be doing, you might have a look at the Euler-Lagrange Equation which is probably the first thing you'll derive in the class.

$\endgroup$
1
  • $\begingroup$ thanks for your detailed answer. $\endgroup$ Commented Jan 31, 2014 at 20:33

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.