Laplace's Equation and Poisson's Equation
Vložit
- čas přidán 2. 06. 2024
- Laplace's equation is one of the most important partial differential equations in all of physics. It is the basis of potential flow and many other phenomena. When forced, it becomes the Poisson equation.
@eigensteve on Twitter
eigensteve.com
databookuw.com
This video was produced at the University of Washington
%%% CHAPTERS %%%
0:00 Overview and Recap of Partial Differential Equations
4:40 Laplace's Equation
6:28 Examples of Laplace's Equation
15:26 Poisson's Equation: Laplace's Equation with Forcing - Věda a technologie
I am an applied math grad student in my final semester. I do feel like I have so much to learn and I still have no idea where all of this will take me . I feel like there are a lot more questions in my universe than there are answers. However, when it comes to you, Dr. Brunton, I don't have any questions. I come back to your instruction time and time again. You have helped me so much see the engineering and data science applications of the mathematics that I have studied. I love that-- what can I do with this?! You always answer that question for me. I appreciate you and the time you take to be "so excited" and how much you "love" literally everything about every single subject you teach! Thank you for your contributions to science and to academics and most selfishly to my journey!
“Naiver Stokes.. those are the PDEs for momentum conservation in a fluid.” - that is such an excellent and simple way of explaining something I, as of only a few minutes ago, could not explain.
Actually the first one is an equation describing conservation of mass, the second one is for the conservation of momentum
Please consider publishing the associates notes to this series as a single repository for quick access and review.
Thank for your consideration.
Beautiful equations ❤️ thanks for the video :)
Brilliantly explained! The provides us a good background before delving into the subject further!
Clicked subscribe! Looking to learn more from you
The Laplace equation also comes into play where you have a flow of global constant vorticity or vorticity which is purely a function of depth.
Hi Steve, is there any possibility of doing a post doc with you?
Much, much appreciated
How potentials are near or on the masses or charges...thanks
Excellent. Thx.
Isnt gravitational potential the potential energy divided my unit mass, so V = -MG/r. Im going off memory so I may be wrong.
Could you do more derivations based on first principles?
Thank you for these videos.
i can't share the link with you, here. Automatic cleaning of the comment section, maybe? I'll try again later
lmth niam/snoitauqEecnalaB/MC/scisyhP/tset/gro atsivretla scisab
Let's try this. Reverse the string and replace white spaces with dots
The demonstration, that the divergence of the field of the gravitation force is null, is so beautiful, so satisfying !
There are two terms :
1. As we live in a 3D space then the dilatation (or divergence) is 3 (1+1+1),
2. but, as we all know, the force is inversely proportionnal to the distance to the square. When a power is at the denominator, then the derivative gives a coefficient : minus the power : -2.
But the trick is that we have to, as Leibnitz said, always keep our eyes in the center of the planet. So we have to see things as a radius : 1/d² = arrow(r)/r³.
And we have the term -3 (the shrink or the convergence) !!
And 3 (as 3D space) + (-3) (as the derivative of a term to the power 3 at the denominator) = 0
Amazing result; So beautiful ! Why are mathematics so adequate to describe the Nature ! It's incredible !
Can you explain this to me like in WIRED's 5 levels of difficulty videos? P.S: I'm level 1 Child😅
@@unknownstoneageman81 When you throw a small stone into the water, you can see a cicrcle expanding or growing. The surface of a lake is a 2D space. So what you see is : +2 as 2 dimensions.
But as you know, the perimeter of a circle is 2.pi.r. r being the radius. So the amplitude of the little wave of the circle decreases as 1/r. But you have to think about it as a vector from the center of the circle and not as a scalar : 1/r = arrow(r)/r².
So +2 -2 = 0
This is the same for the gravitation in 3D space instead of 2 for the stone into a lake. So the next time, when you will throw a little stone in the sea or lake, look up to the sky and realize that the sun acts on our planet as the same as the circle that you see propagating at the surface of the water ...
@@lioneloddo Thankssss. But what do you mean by 1/r tho. Can you describe it visually for me please?
@@lioneloddo Please?
As a French, I am particularly touched by this video about France and our great and famous French mathematicians...
French mathematicians basically built most of math.
One of them is a bit fishy, but still... ;)
@@maciejmanna9246 What do you mean?! Is it a joke about poisson = fish?!
Amazing
1:20 Solely because of the Leibniz's assumption on the continuity in all physical process.
Too good
Great! Love the phrase "mathematical architecture"!
Poisson sounds "fishy" ! Great. twist on French words! 😂
GG to everybody taking PDE class
Vector calculus in a nutshell:)
ToE solution:
The Theory of Everything, as to not be contradictory, would mean Theory of the Fundamentals of Everything.
Could you imagine the Theory of the Specifics of Everything? Haha
I contend Gottfried Leibniz was correct about the fundamentals of our contingent universe and he just lacked 2022 verbiage/common knowledge.
More importantly is that humanity chose Isaac Newton's "real" universe, calculus, gravity, etc. This was a mistake. We need to correct this problem.
I've done my couch 🛋 warrior self's best impression of finishing what Leibniz started (with the intention of destroying what Newton started):
0D = (point); exact location only; non-composite substance.
(How ironic the symbol for Physics is the symbol for Metaphysics/Monad)
1D = line, straight; two points; composite substances
《0D (point) is exact location only; zero size; not a 'thing', not a 'part'; Monad》
"He is the invisible Spirit, of whom it is not right to think of him as a god, or something similar. For he is more than a god, since there is nothing above him, for no one lords it over him. For he does not exist in something inferior to him, since everything exists in him. For it is he who establishes himself. He is eternal, since he does not need anything. For he is total perfection. A being can have a relationship with a God but not the Monad as that would be a contradiction."
- The Apocryphon of John, 180 AD.
Monad (from Greek μονάς monas, "singularity" in turn from μόνος monos, "alone") refers, in cosmogony, to the Supreme Being, divinity or the totality of all things.
The concept was reportedly conceived by the Pythagoreans and may refer variously to a single source acting alone, or to an indivisible origin, or to both.
The concept was later adopted by other philosophers, such as Gottfried Wilhelm Leibniz, who referred to the monad as an elementary particle.
[In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. There is only one 0-tuple, referred to as the empty tuple. An n-tuple is defined inductively using the construction of an ordered pair]
It had a geometric counterpart, which was debated and discussed contemporaneously by the same groups of people. (they didn't have the verbiage/terminology)
1st four dimensions are 0D, 1D, 2D, 3D ✅.
1st four dimensions are not 1D, 2D, 3D, 4D 🚫.
Human consciousness, mathematically, is identical to 4D quaternion algebra with w, x, y, z being "real/necessary" (0D, 1D, 2D, 3D) and i, j, k being "imaginary/contingent" (1D xi, 2D yj, 3D zk).
1D-9D 'contingent' universe has "conscious lifeforms" (1D xi, 2D yj, 3D zk)..."turning" 'time'. We're versors "turners", "to turn". Humanity is (3D zk).
"Turn" to what, you might ask. 5D is the center of 1D-9D. The breadth (space-time). All things and parts are drawn to the center, the whole.
[Contingent Universe]:
3 sets of 3 dimensions:
(1D-3D/4D-6D/7D-9D)
The illusory middle set (4D, 5D, 6D) is temporal. Id imagine we create this middle temporal set similar to a dimensional Venn Diagram with polarized lenses that we "turn" by being conscious.
Which requires energy. 3D height Symmetry with 9D absorption is why we are "consumers", we must consume, i.e. calories, and sleep, to continue "to turn" 'time' (be alive).
1D-3D set/7D-9D set overlap creating the temporal illusion of 4D-6D set.
1D, 2D, 3D = spatial composite (line, width, height)
4D, 5D, 6D = temporal illusory (length, breadth, depth)
7D, 8D, 9D = spectra energies (continuous, emission, absorption)
Symmetry/entanglement:
1D, 4D, 7D line, length, continuous
2D, 5D, 8D width, breadth, emission
3D, 6D, 9D height, depth, absorption
Conclusion: Humanity needs to immediately swap from "Newton" to "Leibniz".
Newton: "wait... Leibniz is Humanity's true universal genius?"
Leibniz (behind Newton w/🔫):
"always have been."
are you left-handed sir
Scalar gravity
little fishy :)
❤️
ToE (relevant information):
According to theoretical physicist Carlo Rovelli, time is an illusion: our naive perception of its flow doesn't correspond to physical reality. Indeed, as Rovelli argues in The Order of Time, much more is illusory, including Isaac Newton's picture of a universally ticking clock.
Does time exist without space?
Time 'is' as space 'is' - part of a reference frame in which in ordered sequence you can touch, throw and eat apples.
Time cannot exist without space and the existence of time does require energy.
Time, then, has three levels, according to Leibniz:
(i) the atemporality or eternality of God;
(ii) the continuous immanent becoming-itself of the monad as entelechy;
(iii) time as the external framework of a chronology of “nows”
The difference between (ii) and (iii) is made clear by the account of the internal principle of change.
The real difference between the necessary being of God and the contingent, created finitude of a human being is the difference between (i) and (ii).
4D = architecture (structure)
5D = design (solution)
Our universal constants have convoluted answers. Leibniz's Law of Sufficient Reason fixes these in a day.
FUNDAMENTALS > specifics.
Our calculus is incorrect (Leibniz > Newton):
What is the difference between Newton and Leibniz calculus?
Newton's calculus is about functions.
Leibniz's calculus is about relations defined by constraints.
In Newton's calculus, there is (what would now be called) a limit built into every operation.
In Leibniz's calculus, the limit is a separate operation.
0D = (point)/NOT a 1D point:
[Math; Geometry]
A point is a 0-dimensional mathematical object which can be specified in -dimensional space using an n-tuple ( , , ..., ) consisting of. coordinates. In dimensions greater than or equal to two, points are sometimes considered synonymous with vectors and so points in n-dimensional space are sometimes called n-vectors.
[Math; 4D quaternion algebra]
A quaternion is a 4-tuple, which is a more concise representation than a rotation matrix. Its geo- metric meaning is also more obvious as the rotation axis and angle can be trivially recovered.
In mathematics, a versor is a quaternion of norm one (a unit quaternion). The word is derived from Latin versare = "to turn" with the suffix -or forming a noun from the verb (i.e. versor = "the turner"). It was introduced by William Rowan Hamilton in the context of his quaternion theory.
How do you make a quaternion?
You can create an N-by-1 quaternion array by specifying an N-by-3 array of Euler angles in radians or degrees. Use the euler syntax to create a scalar quaternion using a 1-by-3 vector of Euler angles in radians.
[Biology]
Points, conjugate. (Science; Microscopy) The pair of points on the principal axis of a mirror or lens so located that light emitted from either point will be focused at the other. Related points in the object and image are located optically so that one is the image of the other.
(See: polarizing element).
I wonder u can write in reverse way so fluently? Aside from the great content of course