
Title:

Series: Advent of Mathematical Symbols

YouTubeTitle: Advent of Mathematical Symbols  Part 1

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Timestamps

Subtitle in English
1 00:00:00,457 –> 00:00:03,800 The mathematical symbol of today is the Kronecker delta.
2 00:00:03,1000 –> 00:00:07,031 Written as lower case delta_i_j.
3 00:00:07,700 –> 00:00:10,438 And the definition is not complicated at all.
4 00:00:10,439 –> 00:00:12,639 It’s either 1 or 0.
5 00:00:13,786 –> 00:00:17,591 Indeed it’s only 1, when i = j
6 00:00:18,557 –> 00:00:21,582 and in all other cases it’s equal to 0.
7 00:00:22,500 –> 00:00:26,713 This means that you can use the Kronecker delta for a lot of possibilities
8 00:00:26,913 –> 00:00:29,878 So for a lot of different indices “i” and “j”.
9 00:00:30,571 –> 00:00:35,305 The only thing that is required is that equality of the indices make sense.
10 00:00:36,471 –> 00:00:40,561 In fact most of the time “i” and “j” represent integers.
11 00:00:41,500 –> 00:00:47,284 For example when you write delta_1_2, it just stands for the number 0.
12 00:00:48,286 –> 00:00:52,821 On the other hand delta_5_5 stands for the number 1.
13 00:00:53,614 –> 00:00:57,214 So you see the kronecker delta is a very simple symbol,
14 00:00:57,414 –> 00:01:02,137 but nevertheless it can be very helpful when you calculate with sums, for example.
15 00:01:02,757 –> 00:01:09,732 This means in the case you have a complicated sum, where kronecker delta is involved usually you can simplify it.
16 00:01:10,171 –> 00:01:13,284 So for this double sum from 1 to 5,
17 00:01:13,484 –> 00:01:15,443 where delta_i_j is involved.
18 00:01:15,686 –> 00:01:17,906 Most of the terms are 0.
19 00:01:18,557 –> 00:01:22,415 Therefore what comes out here is simply the number 5.
20 00:01:23,543 –> 00:01:26,197 Ok, that’s the Kronecker delta.
21 00:01:26,397 –> 00:01:27,615 Thanks for listening.