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I find it is hard to catch the current, sometimes it is just the picture as its support set (if I do not miss it). What is the heart idea of the current? What are the benefits to introduce such an odd concept which make the theory more and more complicated?

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I voted to close as not constructive. I suggest that you read some of the standard references first before voicing such uninformed opinions. A highly readable source with lots of references is Morgan's Geometric measure theory: a beginner's guide. – t.b. Aug 28 '11 at 6:54
In other words: If you want an explanation of what currents are and what their benefits are then why don't you simply ask for that? Why do you need to deem it an "odd concept"? How would you prove compactness theorems without them? Why do you think that they "make the theory more and more complicated"? – t.b. Aug 28 '11 at 7:01
In my view currents are central unifying notion of $20^{\rm th}$ century differential geometry. Unfortunately they have not made their way into the mainstream math curriculum. Therefore it is highly understandable when someone is asking for a link to a pedestrian introduction to these animals. – Christian Blatter Aug 28 '11 at 17:53
Yes,I am reading the book "Geometric measure theory: a beginner's guide".But that is too strange,very dislike the Riemainn geometry or the measure theory. – Strongart Aug 31 '11 at 5:32
Still interested in an answer? – user24894 Feb 13 '12 at 5:18

In case you are still interested in an answer, I encourage you to look at Leon Simon's notes. The idea of currents, is to serve as a far reaching generalization of oriented submanifolds. They were introduced as part of the fairly long and involved program that went into solving the Plateau's problem: such a minimization problem would require three important ingredients: a notion of convergence (topology), admissibility of the "limit", and finally lower semicontinuity. It turns out that these three are in some sense opposing forces, and currents provide the right language to meet all these three ingredients.

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