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The Poisson equation has many uses in physics... so we'll be understanding the basics of the mathematics behind it, and then applying it to the study of classical gravitation!

#physics #poissonequation

We begin by understanding the nabla or del operator, which is a vector of partial derivatives in the x, y, and z directions (since we are working in 3 dimensions).

We then see that the Poisson equation contains what looks like del squared. So how do we square the del operator? Well, del squared is just a notation used to represent del (dot) del, the scalar product between two del operators. Another way to think about this is the divergence of the gradient of some scalar field. Importantly though, del squared is actually the sum of the second partial derivatives of our function, in the x, y, and z directions. And this quantity must be equal to some function (which we have called phi) as part of the Poisson equation.

With this basic mathematical understanding, we then look at how Poisson's equation is used in physics. We begin by recalling Gauss' Law (in integral form first, as this is more intuitive). We then consider the Earth's gravitational field, and understand the meaning of each symbol in Gauss' Law. We see that we can choose an arbitrary closed surface that entirely encapsulates the Earth, break it up into infinitely many area elements, and then study how the gravitational field passes through each of these area elements. We add up the contribution from each of these small areas in order to complete our understanding of Gauss' Law.

But Gauss' Law can be written in differential form too, which relates much more closely to Poisson's equation. Except in the differential form, we have del (dot) g where g is the gravitational field. Another way to think of this is the divergence of the gravitational field. At which point we can understand that gravity is a conservative force. This means that any object moved from one point to another in a gravitational field, will have the same amount of work done on it, no matter what path it takes from its original position to its final position. Additionally, no work is done in moving an object from one point back to that same point, no matter the path taken.

Mathematically, this translates to the fact that the curl of the gravitational field must be equal to zero. And there is a mathematical identity that tells us that the curl of the gradient of any scalar function must be zero, meaning we can write the gravitational field as the gradient of a scalar field V. This scalar field happens to be known as the "gravitational potential", and is an important quantity in understanding gravitation. Moreover, when we substitute this fact into the differential form of Gauss' Law, we find that we are working with Poisson's equation for Gravitation!

Check out these resources:
https://en.wikipedia.org/wiki/Partial...
https://en.wikipedia.org/wiki/Gauss%2...

Timestamps:
0:00 - Intro, thanks for voting in my community poll!
0:28 - The general from of the Poisson Equation (and the form used for Gravitation)
0:55 - The nabla / del operator: vector of partial derivatives
3:19 - Squaring nabla: the Laplace operator, and finding the scalar product between vectors
4:40 - Thanks to Skillshare for sponsoring this video, check out a free trial of Skillshare Premium at the first link in the description below!
5:41 - Gauss' Law for Gravitation, as understood for the gravitational field of the Earth
9:15 - The Differential Form and the Conservative Gravitational Field
10:48 - The Poisson Equation for Gravitation

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