Physics major here in my final year rn, and dude this changed my perspective on vector calculus so much, I love how you encourage a geometric way of thinking. Can't wait for more in this series!
Amazing video!! Just an insight to those taking multivariable calc this year(including me!!): a line integral is similar to a normal integral- when comparing the FToC to the FToLI, when you take a integral of small rate of changes, aka the derivative f'(x), it gives you the total change, f(b)-f(a). Similarly, in vector calculus, you can think of each vector in the field containing a weight, or a numeric value, at each point and thus the FToLI states that the integral of weighted rate of changes gives the total weighted length, which is the definition of a line integral! That's the way I like to view them. Anyways, can't wait for the next video!
IMHO, this is remarkably good content on a par with 3Blue1Brown, ParthG, Mahesh Santoy (Floathead Physics/Khan) et al, and in some ways actually better! I've not explored your other content yet but you clearly have a gift for conveying complex mathematics. Thank you!
Also, the gradient has interesting properties: dz = df(x, y) = ∇f ∙ dr. Finally, vector fields are written with capital letters, such as F, G, H, etc. Electric fields are represented as E, magnetic fields as B, and velocity fields as V. There exists divergence, which measures how the vector field is coming out, which is equal to ∇ ∙ F. When the divergence is positive, it becomes a "source", and if the divergence is negative, it becomes a "sink". This is crucial in determining "fixed points". A fixed point is a point on a path in a vector field that when a mass is placed on the path, it doesn't get influenced by the vector field. They occur when the vector field is normal to the path. Consider Earth's gravitational field G = -10k m/s^2 (or -22.37j MPH in America, as in America, the y-axis goes up and down and the z-axis goes forward and back, which makes south positive Z rather than metric, which makes the z-axis goes up and down, making north positive Y). When a surface has a z-component of gradient that is equal to 0, the object will not be affected by gravity, so it is a fixed point in said vector field. There are 2 types of fixed points: stable and unstable. A stable fixed point is one that when an object is pulled off, it is reattracted back to the point, whereas an unstable fixed point is one that when an object is nudged by the tiniest amount, it will be repelled away from the fixed point. If the divergence is positive, the fixed point will be unstable, and if it is negative, the fixed point will be stable. Finally, the curl of a vector field defines how much it is rotating, which produces a bivector field ∇ ^ F.
you should do a whole series on calculus, probability, from general view to more practical examples, if you need help I could help you edit and animating on 3d
Hi, Any chemical or liquid combination can create a bright white light transparent crystals and that continues to glow for upto 1 month or more. Is it possible to make it ?
I'm glad you asked this question, it's made me think quite a bit ... I like to use it as a way to contextualize vectors, for me it always just made the most sense since we draw vectors as arrows, but it was only after I took calc 3 that I realized there's a deeper connection. Of course, the object of a vector doesn't have any units inherently associated with it, but I also don't think time is necessarily important here. The "movement" I'm really thinking of here is movement along a curve or surface, and having vectors that represent that movement is precisely what gradient vector fields are-telling you how much a function output will change if you move in a certain direction in its input space at a given point. This again is not necessarily related to the concept of an object moving in space and time, or having a velocity (though it definitely can be in appropriate physical contexts). Hope that makes more sense. Thanks for the question, I will be sure to talk about this in my next video!
Physics major here in my final year rn, and dude this changed my perspective on vector calculus so much, I love how you encourage a geometric way of thinking. Can't wait for more in this series!
Amazing video!! Just an insight to those taking multivariable calc this year(including me!!): a line integral is similar to a normal integral- when comparing the FToC to the FToLI, when you take a integral of small rate of changes, aka the derivative f'(x), it gives you the total change, f(b)-f(a). Similarly, in vector calculus, you can think of each vector in the field containing a weight, or a numeric value, at each point and thus the FToLI states that the integral of weighted rate of changes gives the total weighted length, which is the definition of a line integral! That's the way I like to view them. Anyways, can't wait for the next video!
Thanks for the great insight! This is a cool way of thinking about it
you explain it soo good that i only need to watch the video twice.
but pls more and harder examples… just to go sure
to be fair you have to watch it with very much attention
IMHO, this is remarkably good content on a par with 3Blue1Brown, ParthG, Mahesh Santoy (Floathead Physics/Khan) et al, and in some ways actually better! I've not explored your other content yet but you clearly have a gift for conveying complex mathematics. Thank you!
No way I just watched your other calculus video earlier this week! Glad to see another upload!
Also, the gradient has interesting properties:
dz = df(x, y) = ∇f ∙ dr.
Finally, vector fields are written with capital letters, such as F, G, H, etc. Electric fields are represented as E, magnetic fields as B, and velocity fields as V. There exists divergence, which measures how the vector field is coming out, which is equal to ∇ ∙ F. When the divergence is positive, it becomes a "source", and if the divergence is negative, it becomes a "sink". This is crucial in determining "fixed points". A fixed point is a point on a path in a vector field that when a mass is placed on the path, it doesn't get influenced by the vector field. They occur when the vector field is normal to the path. Consider Earth's gravitational field G = -10k m/s^2 (or -22.37j MPH in America, as in America, the y-axis goes up and down and the z-axis goes forward and back, which makes south positive Z rather than metric, which makes the z-axis goes up and down, making north positive Y). When a surface has a z-component of gradient that is equal to 0, the object will not be affected by gravity, so it is a fixed point in said vector field. There are 2 types of fixed points: stable and unstable. A stable fixed point is one that when an object is pulled off, it is reattracted back to the point, whereas an unstable fixed point is one that when an object is nudged by the tiniest amount, it will be repelled away from the fixed point. If the divergence is positive, the fixed point will be unstable, and if it is negative, the fixed point will be stable. Finally, the curl of a vector field defines how much it is rotating, which produces a bivector field ∇ ^ F.
After decades of education, I have realized that Mathematics is all about modelling seemingly complicated universe with weird symbols.
Superbly explained such a complex topic with el' an, kudos to you 👍👍👍
You should have talked about scalar fields, makes much more sens in vector calculus to see multivariable functions as fields
you should do a whole series on calculus, probability, from general view to more practical examples, if you need help I could help you edit and animating on 3d
Hi, can you please provide link for website you used to generate the vector fields
great review😇
is a parameterization the same or different from the original equation? For instance, y=x^2 is not ti+ t^2j, it is ti-t^2j.
Hi, Any chemical or liquid combination can create a bright white light transparent crystals and that continues to glow for upto 1 month or more. Is it possible to make it ?
17:03 i saw that
ı need mkre
Why can you think of vectors as movement? Vectors have magnitude and direction. There's no dimension of time there.
I'm glad you asked this question, it's made me think quite a bit ... I like to use it as a way to contextualize vectors, for me it always just made the most sense since we draw vectors as arrows, but it was only after I took calc 3 that I realized there's a deeper connection. Of course, the object of a vector doesn't have any units inherently associated with it, but I also don't think time is necessarily important here. The "movement" I'm really thinking of here is movement along a curve or surface, and having vectors that represent that movement is precisely what gradient vector fields are-telling you how much a function output will change if you move in a certain direction in its input space at a given point. This again is not necessarily related to the concept of an object moving in space and time, or having a velocity (though it definitely can be in appropriate physical contexts). Hope that makes more sense. Thanks for the question, I will be sure to talk about this in my next video!
DO AS MANY ADS AS YOU WISH I WILL WATCH THE NEXT VIDEO!!!