• Title: D’Alembert Operator

• YouTube-Title: Advent of Mathematical Symbols - Part 29 - D’Alembert Operator

• Bright video: https://youtu.be/ZooZnE2hH5c

• Dark video: https://youtu.be/em_oOvZ-JM0

• Timestamps
• Subtitle in English

1 00:00:00,514 –> 00:00:03,720 Hello and welcome back to the next mathematical symbol.

2 00:00:03,729 –> 00:00:07,501 Which is the d’Alembert operator denoted by such a box.

3 00:00:07,701 –> 00:00:12,747 You might already know, mathematicians often use this box to denote the end of a proof,

4 00:00:12,947 –> 00:00:19,026 but here we will talk about another usage of this symbol. Namely we talk about the differential operator.

5 00:00:19,514 –> 00:00:24,769 Indeed, this d’Alembert operator is often used for applications in physics.

6 00:00:24,986 –> 00:00:32,489 More precisely it’s often used as a differentiable operator when we have 3 dimensions in space and 1 dimension in time,

7 00:00:33,014 –> 00:00:37,867 because then we are able to combine some seconde order derivatives.

8 00:00:38,700 –> 00:00:45,704 First we have the one with respect to the time variable t minus the ones with respect to the spatial variables

9 00:00:46,214 –> 00:00:51,400 and they are simply put into a Laplacian, the three dimensional Laplace operator.

10 00:00:51,929 –> 00:00:56,499 So there you should know, this is simply the sum of 3 partial derivatives

11 00:00:57,114 –> 00:01:02,313 and you see, usually we call the 3 dimensions in space x_1, x_2, x_3.

12 00:01:02,957 –> 00:01:07,011 Hence what happens here is that we connect space, time.

13 00:01:07,211 –> 00:01:11,902 So the d’Alembert operator is used when you want to describe something in space-time.

14 00:01:12,371 –> 00:01:20,459 For this reason the units where we measure space and time, should coincide and therefore the speed of light should occur somewhere.

15 00:01:20,886 –> 00:01:24,870 Indeed often we just put it in front of the time derivative.

16 00:01:25,300 –> 00:01:30,687 So you find 1 over c^2 there, where c stands for the speed of light.

17 00:01:31,157 –> 00:01:37,071 Ok, there we have it. This is the meaning of the box, when we use it as a differential operator

18 00:01:37,271 –> 00:01:41,782 and then I would say, let’s meet in the next video about mathematical symbols.

19 00:01:41,886 –> 00:01:42,800 Bye!

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