We have a function of x and y equal to z, we can make this a function that is of x y and z by subtracting, this new function describes a level surface, that's jargon to me right now, so we elaborate. The first bit of that is surface, well not in english but never mind, a surface is like a plane that is all blobbed up and mushed together in some way. We can level a surface by setting it equal to zero, this is a weird concept frankly, but that 3D surface is a cross section of a 4D figure. The important thing is that we have F(x,y,z)=0 describing this surface. Then we have this gradient thing, what's that? I dunno, but we're gonna BS it and pull it out of thin air without the help of the internet. So V is gonna be my stand in for the gradient symbol... VF(x,y,z)= the partials of each component. What does this mean? Do I really understand partial derivatives is the question here? Well a gradient shows how x, y, and z change with time. It is a vector quantity, and this makes sense because if you pick a point on a surface and vary time, it should change in some direction that could be changing. Say you had this vector that describes how a surface is changing. Where is it going to be at a given point? Well I can only visualize usefully in 2 dimensions, say you want to draw a circle with a vector that describes how it changes in time. This vector has a perpendicular component changing it to go towards the center, and a tangential component that drags it sorta like orbit. I think these are different orders of derivatives though, right? Yeah, velocity is changing with acceleration, so what we're actually looking at for this model is how a tangential vector changes. The gradient then would be orthogonal to the tangent vector running along the surface, but well that's a stupid way to view that, we're going to just say that it is orthogonal to the surface itself, this also isn't a nice surface necessarily, so we resort to partials to make this orthogonality a Boolean equal to true. What do we do with all of this? Well we've got a gradient(Orthogonal vector) and we've got a Surface F(x,y,z), if we can lay this gradient at any point on this surface, we will have the Normal and a point to describe a plane that is kissing the surface, that's a tangent plane. Hey, and that gradient is orthogonal to this surface. Let's say a line is the Normal line to this surface if it has the same direction as this plane and passes through the point where the tangent plane and the surface F(x, y, z) kiss! Okay, now I think I have justified it all in my brain. Continue.
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
thank you so much! this seemed super complicated but your clear explanation makes it seem so simple
YOU ARE THE GREATEST OF ALL TIMES INDEED!!!!!!!
YOUR VIDEOS ARE ALWAYS STRAIGHT TO THE POINT, I'm learning a lot from you Prof.
awesome!
Thank you so much! you are one of the best channels for short but great videos teaching math and calculus.
This is so well explained! Kudos to an awesome teacher
Thank you!
amazing video you made the concept extremely simple to understand! Thank you
you are welcome I am so happy this video helped someone:)
Thank You So Much! Well Explained ❤️
Thank you so much!!! You just saved me 5 pts from my homework xD
math 28?
section? hAHAHAHA
@@von2728 elbi ka?!?
@@0001-exe yessss hahahaha
@@von2728 oh my gosh LEZGOOOO hahahahahah
Very useful Sir, thank you so much ❤
hey just a question at 12:13 (bottom left) what happens if C was 0 for the symmetric equation would you write z=4 instead or what?
This was nice, thank you
👍
thank you ! this was really helpful. :)
Do you ever think of doing book reviews on calculus 3?
Thanks alot
thx bruh
np:)
nice
Thx
Okay, time to piece things together in a way only the one speaking and the all knowing ones can possibly understand!
We have a function of x and y equal to z, we can make this a function that is of x y and z by subtracting, this new function describes a level surface, that's jargon to me right now, so we elaborate.
The first bit of that is surface, well not in english but never mind, a surface is like a plane that is all blobbed up and mushed together in some way. We can level a surface by setting it equal to zero, this is a weird concept frankly, but that 3D surface is a cross section of a 4D figure. The important thing is that we have F(x,y,z)=0 describing this surface.
Then we have this gradient thing, what's that? I dunno, but we're gonna BS it and pull it out of thin air without the help of the internet. So V is gonna be my stand in for the gradient symbol...
VF(x,y,z)= the partials of each component. What does this mean? Do I really understand partial derivatives is the question here? Well a gradient shows how x, y, and z change with time. It is a vector quantity, and this makes sense because if you pick a point on a surface and vary time, it should change in some direction that could be changing.
Say you had this vector that describes how a surface is changing. Where is it going to be at a given point? Well I can only visualize usefully in 2 dimensions, say you want to draw a circle with a vector that describes how it changes in time. This vector has a perpendicular component changing it to go towards the center, and a tangential component that drags it sorta like orbit. I think these are different orders of derivatives though, right? Yeah, velocity is changing with acceleration, so what we're actually looking at for this model is how a tangential vector changes. The gradient then would be orthogonal to the tangent vector running along the surface, but well that's a stupid way to view that, we're going to just say that it is orthogonal to the surface itself, this also isn't a nice surface necessarily, so we resort to partials to make this orthogonality a Boolean equal to true.
What do we do with all of this? Well we've got a gradient(Orthogonal vector) and we've got a Surface F(x,y,z), if we can lay this gradient at any point on this surface, we will have the Normal and a point to describe a plane that is kissing the surface, that's a tangent plane. Hey, and that gradient is orthogonal to this surface. Let's say a line is the Normal line to this surface if it has the same direction as this plane and passes through the point where the tangent plane and the surface F(x, y, z) kiss!
Okay, now I think I have justified it all in my brain. Continue.
Genius😃
@@TheMathSorcerer
Isn't it just restating the stuff I learned in my own words? Every textbook and teacher expects a genius out of every child then.
Ya but you do it so so well
Most people can't do what you just did man, you are good!!