- Telegram - YouTube - Summary ▶️ Watch on 3Speak - Telegram - YouTube - Summary - Hive Notes - Sections playlist - MES Links --- In this video I go over a quick math review of the terms involved i...
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So mathematical review now, before we start deriving a lot of the equations. So Laplace's equation is a second-order partial differential equation named after Pierre Simon Laplace, who first studied it in 1786. A partial differential equation, PDE, right here, is an equation which involves a multi-variable function in one or more of its partial derivatives. A multi-variable function is a function with more than two arguments, such as the volume of a cone. So, for example, volume is equal to, well, it's dependent on the area of, yeah, this is the area, this is cross-section right here, this area of the base, and H, the height of the cone. So basically, you're going to have a low volume, and so on. So that's a multivariable function. So volume is equal to 1 divided by 3, area, of the base times by the height. And here is another figure. So where R is the radius of the base of the cone, and H is the, oh, actually, you're right it as well. This is pi r squared is just area. That's just pi r squared is the area there. So volume equals 1 over 3, so 1 third pi r squared times H, where R is the radius of the base of the cone, and H is the perpendicular height of the cone. Going further, so a partial derivative is a derivative of a multi-variable function with respect to one of those variables with the others held constants. For example, you could take the derivative of this volume right here in terms of, you could do the partial derivative of V of, in terms of H, for example. So basically, this would just look at the rate of change of this height here, ignoring this one here. So you set it. you can go at whatever value and then basically the height changing. Yeah, yeah, this is the volume changing with height and so on, so that'd be pretty interesting. It would be something like this. So you'd be, like, changing the volume, what's the rate of change of that volume, just by doing that? But then if you include the height, and R, you're going to get something like this. And then, so basically, the thing about the component, so basically this would be the rate, the acceleration going up, or the rate change and then this one is v over r like that and the top one is h so anyways so the partial derivative of a function f xy dot dot with respect to the x direction is denoted by any of the following way so do go f of x x x prime go partial x f d capital d x f d x f and this one i'm most used to is basically this or this So partial F, yeah, partial, you have this partial sign. It's like that, that's an interesting symbol. What's called? And, yeah, actually, I just quickly asked Leo AI, this is actually, it's also called Dell, so partial or Dell. Anyways, so it could be thought as the rate change of the function in the X direction. So basically, again, similar to this one, this would be partial v, partial H would be in this upwards direction. All right, fascinating. So for a function F, X, Y, and Z of three variables, Laplace's equation is, so now we're using the F, aside of the U, as I did earlier in the video. So, partial. the second partial derivative in terms of x of f plus the second partial derivative of f in terms of y plus the second partial derivative of f in terms of z equal and then all this equals to zero hence there's no rate change in this respect so that's why the solution is harmonic or in harmony with nature so a function that obeys this equation is called a harmonic function and the Laplace equation can be written using the Laplace operator or Laplaceian So you can do this, this is the triangle there, delta, or the divergence operator, so now you have the upside down one, or of the gradient F. So this divergence one is this gradient thingy, upside down triangle, dot, or I think it's called NABLA, and then the gradient is, this is the gradient, so if you do the dot, this is the divergence operator times by the gradient there. So what this means is, So this basically, so here's the Laplacian or the Laplace operator delta F equals to divergence squared f, and then it's just equal to this dot this one, and then this all equals zero. And basically you have to get to this all just equals to that. And the NABLA symbol, this upside-down triangle, written as an upside-down triangle and pronounced DEL. This one's also pronounced DEL. This was DEL. All right, anyways, I'll just say that partial differential operator, and this just equals to basically the partial derivative of each term. So that is what that is. The gradient, yeah, that's this part right here. So you can just do this right here. Delta F is just the first partial derivative of every single term. So you just remove these. and yeah, that'll be the Dell. And now you do the second del, second divergence operator. The gradient is the vector, yeah, this is the gradient is the vector field or vector function, delta F, or del F, whose value at a given point gives a direction and rate of fastest increase. Again, this is the, yeah, this is the velocity or the rate of change. And then this would be the acceleration. That's acceleration. This one is just the velocity or the rate of change. In coordinate free terms, the gradient of a function, F of R, may be defined by, yes, now we're dealing with the gradient, so DF just equals to, this is the Dellf .D.R, where DF is a total infinitesimal change of F for an infinitesimal displacement of R and can be seen to be maximum when DR is in direction of the gradient F. There. So this is pretty fascinating stuff here. This is a dot there. Okay, so this is the figure of the gradient represented by the blue arrows denotes the direction of the greatest change of a scalar function. And yes, that's the greatest change there. The values of the function are represented in gray scale and an increase in value from white, low to dark high. So high value, and this is low value there. There's a low value there. And so it's just showing the direction of where the function is changing. This one is moving. Yeah, moving towards a dot there, to the right, I'm to the left. And this is used to denote, for example, magnetic or electric field. This one, for example, electric field around a charged object, and it's pointing inward, so that's pretty faster. So in 3D coordinates, the divergence is just the sum of partial derivatives and uses the dot-product notation as a mnemonic device, basically helps to aid in and remember it. So, for example, div f, is just equal to, I think it would be similar to all this. Yeah, anyways, that was for another day. Yeah, anyways, go back here. So divf equals to the divergence. Dotf, and this just equals to, yep, the partial derivatives right there. And again, it's using dot product where you just multiply each of the components and add them up. So this is a partial, partial derivative of X, in terms of X, partial derivative, in terms of Y, partial. partial derivative in terms of z dot this is the fx f y fz components and then basically what you do is multiply this by this, this by this, and then this by this, and add them all up so that's just a way to multiply it in terms of just take a partial derivative so that's equals this but in this case we're doing this of the gradient there so we already have the gradient, so we already have this of F, and it would do it twice. So basically you're just doing it twice. So it's pretty cool stuff. Here's some more interesting stuff. The divergence of two vector fields in two dimensions. So if you have, let's see how this is, the partial derivative of this is the X term, where it's greater than zero, it would be going outwards like that. Interesting. This is where the X component is going outwards. Yeah, this is in two dimensions. So the x and y are both expanding. So the rate change is both expanding and that the, yeah, and basically the divergence or the, this is the gradient. I think that'll be called gradient. Let's go back here. Yeah, this is divergence operator. So go back here. Yeah, you're going back here. So basically, if the divergence of this is the rate change is increasing in the X direction and Y direction, and then you also have the divergence dot there is also increasing. So basically that just all means, it's all increasing. So it's expanding, but if it's all negative, you get this. Yes, it's all going decreasing, decreasing, so on. And then if it's all equal, you basically have no divergence. It's all equal. In other words, you just have straight lines. So that's pretty fascinating stuff. So these ones, you can think of it as charges, this would be minus, it would be plus, et cetera. And this is zero. That's not a charger. Anyways, that'll be for another day, but I just want to wrap my head around that. Right, fastening stuff.