That's exactly the reason why this operator is so important in physics. If any equation containing differential operators represents a "fondamental", general physical phenomenon, then it must respect the rotation invariance. That's why the main equations of classical physics, wave physics, quantum mechanics... Contain a laplacian combined with time operators, although in special relativity, that combination must be done with respect to lorentz invariance, meaning it can only appear in the form of a D'Alembert operator. Anyway, even in those cases, the space dependance is mainly a laplace operator
It’s even scary the fact that yesterday I was working a solid mechanics exercise and I figured that the whole topic was just working with rotating space, and then this video comes along 😂 thanks for explaining this deeper
In RF and microwaves, we use the invariance of Laplace’s equation under the Schwartz-Christoffel transformation to solve for the electric field on a transmission line of any geometry. Professor Collins used this property to calculate fringing capacitance! Absolutely amazing!
Nice, try to derive physical conservation laws from translation in space, time and from rotations in space. I mean Noether's theorem from math view. Best regards
i dont really understand why the laplacian is not motivated such that those properties are immediate: for instance if you have an operator that meassures the mean second derivative across all directions wouldnt someone come up with this property the very moment you see that if you always take all directions into account no matter how you rotate you get the same result? the laplacian does exactly that, being the trace of the Hessian ...
Ooh! @ 1:37 (just before) is the differentiating of the second column of the X-Y rotation matrix a subtle key to some important mathematical property that distinguishes between the [R(1,1), R(2,1)] & [R(2,1), R(2,2)] columns, where R(m,n) are of course the elements of the 2-D rotation matrix?
Good catch, jason! With the angle measured counterclockwise, the matrix given converts new coordinates (x', y') into old (x, y), not the other way around. It turns out, this doesn't really hurt the argument because... To convert in the other direction requires the inverse matrix, which is the same as the transpose (this is what it means to be orthogonal!) If you want to fix this to make it exactly correct, you either relabel the diagram, swapping the two different pairs of coordinates, or you do the entire calculation with the prime and non-prime coordinates swapped out. Either way, you get the same conclusion because the final equation is symmetric under this exchange!
For the divergence, Laplacian and curl: in order for the operators to remain invariant: the transformation must be affine: the Jacobean J of this transformation is J where J is invertible For all J the form for the divergence operator is invariant The Laplacian operator is invariant iff J is in O(3) The form for the curl operator is invariant iff J is in SO(3)
That's exactly the reason why this operator is so important in physics. If any equation containing differential operators represents a "fondamental", general physical phenomenon, then it must respect the rotation invariance. That's why the main equations of classical physics, wave physics, quantum mechanics... Contain a laplacian combined with time operators, although in special relativity, that combination must be done with respect to lorentz invariance, meaning it can only appear in the form of a D'Alembert operator. Anyway, even in those cases, the space dependance is mainly a laplace operator
Pradow dependence & fundamental. He presentation layer comes before the application later!! (TCP/IP & writing)
Dancing with your dependents can be a depen-dance though :)
You are so happy its awesome! I never had Instructors this motivated :)
Interesting, you made a video on path indepence, now rotation invariance, seems like Noether's symmetry and conservation laws
It’s even scary the fact that yesterday I was working a solid mechanics exercise and I figured that the whole topic was just working with rotating space, and then this video comes along 😂 thanks for explaining this deeper
In RF and microwaves, we use the invariance of Laplace’s equation under the Schwartz-Christoffel transformation to solve for the electric field on a transmission line of any geometry. Professor Collins used this property to calculate fringing capacitance! Absolutely amazing!
I appreciate your help.
Nice, try to derive physical conservation laws from translation in space, time and from rotations in space. I mean Noether's theorem from math view. Best regards
i dont really understand why the laplacian is not motivated such that those properties are immediate: for instance if you have an operator that meassures the mean second derivative across all directions wouldnt someone come up with this property the very moment you see that if you always take all directions into account no matter how you rotate you get the same result? the laplacian does exactly that, being the trace of the Hessian ...
Ooh! @ 1:37 (just before) is the differentiating of the second column of the X-Y rotation matrix a subtle key to some important mathematical property that distinguishes between the [R(1,1), R(2,1)] & [R(2,1), R(2,2)] columns, where R(m,n) are of course the elements of the 2-D rotation matrix?
I think it’s just a coincidence :)
Dr Peyam maybe.... maybe there is a deep connection between differentiation and dimension? New realms await exploration, perhaps?!!
Hey, Dr Peyam! How about to tell about theory of groups?
1:20 Is there any special reason why the second column is differentiation of the first one?
Coincidence :)
The sign of rotation matrix seems incorrect ,the minus sine should be at low left conner not up right conner.
am I right ?
Nope, its right
Good catch, jason! With the angle measured counterclockwise, the matrix given converts new coordinates (x', y') into old (x, y), not the other way around. It turns out, this doesn't really hurt the argument because... To convert in the other direction requires the inverse matrix, which is the same as the transpose (this is what it means to be orthogonal!)
If you want to fix this to make it exactly correct, you either relabel the diagram, swapping the two different pairs of coordinates, or you do the entire calculation with the prime and non-prime coordinates swapped out. Either way, you get the same conclusion because the final equation is symmetric under this exchange!
What about volume bounded by the intersection of 2 cylinders at an angle ? How to solve it ..
Already done
@@drpeyam but sir u have done the cylinders at right angles but not at any angles.
@@drpeyam can u plz do it
Probably not
@@drpeyam so can u do it sir ??
For the divergence, Laplacian and curl: in order for the operators to remain invariant: the transformation must be affine: the Jacobean J of this transformation is J where J is invertible
For all J the form for the divergence operator is invariant
The Laplacian operator is invariant iff J is in O(3)
The form for the curl operator is invariant iff J is in SO(3)
Sir please make a video on green's function 🙏
Maybe
When Sir?
@@drpeyam sir plz make video
Sir can you do the same for n dimensions?
No thanks
@@drpeyam please sir I got stuck in this problem
Sorry
@@drpeyam 😭😭
Hey dr payem 🤗
Wow
@@dm_saj1119 wath bro
@@اسلاماسلام-ت2ز5و u love math
Mi tooo
🌺
"U dos equis" 😂😂😂
These details must be worked alongside with on paper or with electronic means. Alphabet soup, otherwise!!