• Title: Closed Line Integral

• YouTube-Title: Advent of Mathematical Symbols - Part 32 - Closed Line Integral

• Bright video: https://youtu.be/2IjrkGOxndo

• Dark video: https://youtu.be/GOQq1FtW0Kw

• Timestamps
• Subtitle in English

1 00:00:00,386 –> 00:00:03,443 Hello and welcome back to the next mathematical symbol.

2 00:00:03,557 –> 00:00:06,738 Which is the integral sign with this circle inside.

3 00:00:07,014 –> 00:00:11,910 So you see, this is a variation from the integral sign, from the last video

4 00:00:12,110 –> 00:00:18,266 and here the circle always tells you that the integration is closed in some sense.

5 00:00:18,466 –> 00:00:23,098 For example often it denotes a so called “closed line integral”.

6 00:00:23,557 –> 00:00:30,022 There we simply have a curve in some space. For example the plane and we integrate along it

7 00:00:30,222 –> 00:00:37,515 and exactly this is what we would call a line integral and most importantly, it has a starting point and an end point

8 00:00:37,971 –> 00:00:44,810 and now this circle here gives the additional information that the start point and the end point coincide.

9 00:00:45,010 –> 00:00:49,873 So a closed integral means that the curve ends where it started.

10 00:00:50,073 –> 00:00:57,103 So indeed the curve could look complicated. It’s just important that we find the end point at the beginning again.

11 00:00:57,500 –> 00:01:05,132 Ok and now it turns out that these integrals occur so often that it’s useful to introduce a new symbol for them.

12 00:01:05,332 –> 00:01:12,233 For example in complex analysis it turns out that this closed line integral carries a lot of information.

13 00:01:12,433 –> 00:01:19,173 You can look at a function 1/z and then it turns out that this closed line integral is not always 0.

14 00:01:19,373 –> 00:01:27,407 Indeed we find that for the circle that goes one time around the origin, this integral is exactly 2pii.

15 00:01:28,057 –> 00:01:31,901 So this is what we can easily calculate in the complex plane