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# Gradient Squared | Laplacian Of A Scalar Or Vector Field | Lecture 20 | Vector Calculus For Engineers

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## Laplacian Of A Scalar Or Vector Field | Lecture 20 | Vector Calculus For Engineers

Transcript:

[MUSIC] You, so in this video, I want to talk about the laplacian. We’ve talked about the gradient, the divergence and the curl already they will use this Del operator, which is a differential operator. The laplacian is written as del squared, which comes from this differential operator dotted with itself. These unit vectors are orthonormal. So this is just the derivatives D squared. DX squared plus d squared dy squared plus d squared. Dz squared so the sum of the second partial derivatives. This this is called the Laplacian operator. It can act on both a scalar field or a vector field, so for a scalar field. We would have del squared. F with BD squared F DX squared plus d squared. F dy squared plus d squared F DZ squared, ok? What about if Del squared? The laplacian acts on a vector field? So for a vector field? We could write down. Squared you how you interpret. This is that Del Squared acts on each component of U. So this would be Del Square U 1 times. I plus del square u 2 times. J plus del squared u 3 times K ok? Why is the laplacian so important? It’s because it shows up in a lot of PDEs. So let me show you some P DS. So the P DS is. This is a partial differential equation. I talked about PDEs in my differential equations for engineers course. Let me give you some three representative. Pde s where you have the laplacian. The first one is simply Del Squared scalar field equal to 0 That’s just called the Laplace equation after the Laplacian and after the mathematician Laplace, the next one would be the wave equation. The wave equation is the second derivative of some scalar times. DT squared and that’s equal to C squared times the laplacian of the scalar. So that’s the wave equation. OK, it’s called the wave equation because this solution is going to be waves. It can govern things like the music that’s made from a guitar string. For instance, it’s also governs electromagnetic waves and, in fact, will derive this equation for the electric field and the magnetic field, the third equation, which I want to present is the diffusion equation, which is a first derivative with respect to time D. Phi DT equals a diffusion constant times. Del Squared Phi. So this is the diffusion equation. OK, so that governs the diffusion of a dye say in some fluids such as water. This is an equation, which I actually solve in my course differential equation for engineers. OK, so let me give you an example, then of the Laplacian acting on a scalar field, rather simple example, what would be laplacian of the scalar field given by X squared plus Y squared plus? Z squared. Okay, very simple. This is just the second derivative with respect to X would be the Y is held constant, so that would be 2x would be the first partial with respect to X and 2 would be the second. Plus, the second partial derivative, with respect to Y is also a 2 plus the second partial derivative of Z squared with respect to Z is also a 2 so this is simply a 6 OK, so let me review in this video. Then I introduce another differential operator. Which is that just the Del Operator operator thoughted with itself. It’s called the Laplacian. It’s the sum of all the second derivatives in Cartesian coordinates. OK, in Cartesian coordinates, it can act on a scalar it can act on a vector and it acts on a vector by acting on these on the components of the vector. Separately, you see a term like this in the navier-stoke’s equations, which is the equations governing fluid motion, it shows up in many other. Pds, including the Laplace equation, wave equation and diffusion equation. It also shows up in the Schrodinger equation for quantum mechanics. OK, and this is an example of how you compute the laplacian given some scalar field. I’m Jeff Jasanoff. Thanks for watching and Ill. See you in the next video.

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