I find it rare to both understand an equation intuitively and how to calculate it after watching a video. You are raising the bar of education everywhere.
Congratulations for your videos, I am now a very old man and when I was younger about 65 years ago, I tried to clarify in my mind the different activities that the few derivatives and associated integrals, contribute to the following set of particular "activities/ functions" they create I could see all this as an Engineering function in my own mind, but never drew my concepts on papers. They seem to have the same building blocks. 1. Cauchy Riemann relations 2. The Grad operator. 3. The curl operator. 4. the Divergence operator. 5 . Green's Curl theorems of circulation 6. Green's Divergent theorem of flux 7. Stoke's Curl theorem involving circulation 8. Divergence theorem involving divergences through volumes, I always thought that students should see the close links there are in how these derivatives are combined to produce their " engineered" activity. dU/dx dU/dy dU/dz dV/dx dV/dy dV/dz dZ/dx dZ/dy dZ/dz and reduced to two dimensions dU/dx dU/dy dV/dx dV/dy
Thank you for this lecture series sir. I have my end semester exams now on Integration in Vector Fields and Multivariable Calculus. You have helped a lot!!!
Its been many years now, but I seem to recall that there are two 'things' about vector fields that, if known, tell us all there is to know about the field. One is where are the sources and sinks , if they exist, located within the field, and the other is where in the field are the spots that might cause a rotation of the field, located. Given the equation of the vector field, if you then run the divergence on it, using the dot product of nabla with the field equation, you will end up with a formula that contains 3 functions of x,y and z [ assuming Cartesian co-ordinates], that are then added algebraically. If you then enter any values of x, y and z into this formula the result with be a scaler number that is either +ve, -ve or zero. If its +ve, its telling you that at that xyz location, the field lines are diverging away from the point and that there is a source of the field there. If its -ve, its telling you that, at that location, the field lines are converging toward the point and that there is a sink of the field there. I find this idea easy to visualize by thinking about a static charge distribution and considering the electric field. If my divergence equation gives me a positive answer, then its telling me that at that xyz location there is a +ve charge ie a source of the field. This idea also explains why the divergence of the magnetic field is zero, since the field lines form closed loops and have no source or sink. The curl of a vector field is, in my opinion, very similar to the angular momentum vector. The curl vectors are just lines you can draw to represent a rotation of the field. They also have no sources or sinks, which, again, explains why the divergence of the curl is zero.
it was so easy, but even tho our prof is great i didn't got the concept, vector calculus is best with all these animations, really thanks for all these efforts!!!!!
So if I got this right, we find the flux which is how much something tends to pass normally past a surface. A pretty formula. Question to self: What would be the boundary and parameterization? We could use spherical coordinates, r*dr*dtheta*dphi and find the boundaries that way.
thanks! Having studied both flux forms of the 2D green theorem, I was wondering why there was something missing in the 3D Kelvin-Stokes Theorem! it only has a 3D curly form , but finally, now found the 3D divergence form (its in the name duh!)
Very good video! One small misprint: The text in small letters entering at 6:23 in the bottom left corresponds to Stokes' Thm., rather than to Divergence Thm.
Masterful graphics and presentation, but there is something that has been gnawing at me in the 1D, 2D and 3D cases: what about flux of a field which does not cross a curve, does not cross a surface, and does not cross an enclosed volume in the normal direction (positive, going out) to the the curve , to the surface or to the enclosed volume? There are an infinite number of directions by which the flux can proceed outwards, of which only one is proceeding outwards (crossing) in the normal direction. How therefore is this total flux, in all directions outwards, calculated?
Excellent lecture. what is the dot product of operators, though. I'm aware of dotting points/vectors, though not operators. And, just curious, what kind of Mathematical Object is the Divergence of a Vector Field?
Imagine a vertical boundary. The flux from left to right is the negative of the flux from right to left. So if we add up both fluxes it would be zero. This is true in the interior. But for a boundary, you only get the one side, not the other.
Why can't the portion along the boundary cancel while what's bounded (inside) can? at ~6:00? Also, would this work if the divergence was negative and there was contraction? Or would it contradict the unit normal vector???
I thought taking the divergence of a vector changed it into a scalar and the gradient of a scale function turns it into a vector. If you’re taking a vector and multiplying it by the del operator wouldn’t that technically be taking the gradient
If you take a vector field and "multiply" it by the del operator, that could be divergence, or that could be curl, depending on which kind of multiplication it is. In the case of divergence, it is an operation analogous to the dot product, where you distribute each differential operator within the del, to each corresponding term of the vector field, and then add them up. Not really multiplication by strict definition, but I get what you mean. In the case of curl, it is an operation analogous to the cross product. Each differential operator operates on a non-corresponding term, in a pattern spelled out by the determinant of the unit vector row, the differential operator row, and the vector field row. The order matters, and it produces a vector.
The gradient is formed when you start with a scalar field, and use the del operator to take its derivatives, and generate a vector with them as its components. This is an operation analogous to multiplying a vector by a scalar.
He's using the M/N/P trio of letters to name the three component functions that define the vector field. It is very common to avoid O as a variable name, so P is what follows. Some books/instructors call them P/Q/R, which was the trio of letters my instructor chose to use. The vector field is given as: F(x, y, z) = Or in another notation F(x, y, z) = M(x, y, z)*ihat + N(x,y,z)*jhat + P(x, y, z)*khat
d/dx is a verb. dM/dx is a noun. d/dx says "take the derivative of the following, with respect to x" dM/dx says "the derivative of M with respect to x" In this case, the derivatives are really partial derivatives, so these would be those funky d's, rather than ordinary d's. It still means approximately the same calculus action as a derivative in general.
There is a book on tensor and there is one equation written in it about the divergence of a vector field in such way...... div(S)=LimV->0 (1/V) { Sn dA..... Where curly brackets are nothing but integration sign over the domain (or surface area). I am unable to understand the limit part and from where this V (volume) comes into the equation. Help me understanding this please!
