-
Title: Laplace Operator
-
Series: Advent of Mathematical Symbols
-
YouTube-Title: Laplace Operator
-
Bright video: https://youtu.be/2yGeeOkv2ys
-
Dark video: https://youtu.be/XBX2Ug4Xvhg
-
Ad-free video: Watch Vimeo video
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: aoms20_sub_eng.srt
-
Timestamps (n/a)
-
Subtitle in English
1 00:00:00,629 –> 00:00:01,997 Hello and welcome.
2 00:00:02,314 –> 00:00:05,571 and the mathematical symbol of today is the laplace operator.
3 00:00:05,643 –> 00:00:07,274 Denoted with a capital Delta.
4 00:00:08,157 –> 00:00:12,594 This famous operator is also often called laplacian.
5 00:00:13,586 –> 00:00:19,853 and indeed there is no single laplace operator. There are a lot of different meanings depending on the context.
6 00:00:21,029 –> 00:00:32,333 However often the first time one sees triangle in mathematics or in physics is as an differential operator on the space R^3 or in general R^n.
7 00:00:33,386 –> 00:00:39,823 and then it simply acts on functions f defined on the space, but with values in R.
8 00:00:40,771 –> 00:00:46,629 Hence laplace f just denotes the new function also defined on R^3.
9 00:00:47,543 –> 00:00:51,904 So here the variable in R^3 is simply denoted by x.
10 00:00:52,671 –> 00:00:55,943 So we have components x_1, x_2 and x_3.
11 00:00:56,786 –> 00:01:01,305 Indeed with respect to these components we look at a partial derivative.
12 00:01:02,200 –> 00:01:08,175 and the important thing to remember here is, for the laplacian we always look at the second derivative.
13 00:01:09,186 –> 00:01:14,699 So the first thing we have here is the second partial derivative with respect to x_1.
14 00:01:15,571 –> 00:01:18,643 and then you see we simply add the next one.
15 00:01:18,800 –> 00:01:22,929 Which is the second partial derivative with respect to the second component.
16 00:01:24,200 –> 00:01:27,740 and then we simply continue adding the second partial derivatives,
17 00:01:27,940 –> 00:01:32,621 until we reach the second partial derivative with respect to the last component.
18 00:01:33,671 –> 00:01:38,543 So therefore please remember here we don’t have mixed second partial derivatives.
19 00:01:39,371 –> 00:01:43,740 However all the other once, the unmixed once, are added.
20 00:01:44,714 –> 00:01:51,298 Therefore now i think you shouldn’t have any problems writing down the laplace operator defined on R^n.
21 00:01:52,229 –> 00:01:56,960 Ok and with this i hope that you learned something today and that i see you next time.
22 00:01:57,160 –> 00:01:57,900 Bye!
-
Quiz Content
Q1: What is the Laplace operator in one dimension, so for a function $f: \mathbb{R} \rightarrow \mathbb{R}$?
A1: $\frac{ \partial^2 f}{\partial x^2}$
A2: $\frac{ \partial^3 f}{\partial x^3}$
A3: $\frac{ \partial f}{\partial x}$
A4: $\frac{ \partial f}{\partial x} + \frac{ \partial^2 f}{\partial x^2}$
Q2: What is the Laplace operator in three dimensions, so for a function $f: \mathbb{R}^3 \rightarrow \mathbb{R}$?
A1: $\frac{ \partial^2 f}{\partial x_1^2} + \frac{ \partial^2 f}{\partial x_2^2} + \frac{ \partial^2 f}{\partial x_3^2} $
A2: $\frac{ \partial^3 f}{\partial x_1^3}$
A3: $\frac{ \partial f}{\partial x_1} + \frac{ \partial^2 f}{\partial x_2^2} + \frac{ \partial^3 f}{\partial x_3^3} $
A4: $\frac{ \partial f}{\partial x_1} + \frac{ \partial f}{\partial x_2} + \frac{ \partial^3 f}{\partial x_3} $
-
Last update: 2024-11
