I think he means vector arguments that don't depend further on a scalar value. Remember that we are doing differential geometry. I suspect he would be fine with a function like a reflection that has a vector argument that depends on time (or some other parameter). Those are the linear transformations from linear algebra, and you can certainly take a derivative with respect to whatever parameter you have used.
Excellent point and I should make another video that clarifies this point. In this course, I use the term "vector" in the narrow sense of directed segment. It is explained in this video: czcams.com/video/N32KI6qoeRA/video.html. Even with that definition, you could conceive functions of vector argument - for example "length" is a function of a vector. So when I say that "there is no such thing as a function of a vector argument", I mean "for the purposes of this narrative, there is no such thing as a function of a vector argument". And that's good news - fewer types of objects makes for a simpler framework.
Of course you need f:R^n->R^m, and of course you can take its directional derivative (supposing it’s differentiable). f’(x_1, …,f_n) in the y (vector in R^n) direction will be a vector in R^m whose i-th component is the dot product . The total derivative will be a linear operator. The subject is comprehensively treated in Apostol’s volume 2.
5:32 I assume you're not talking about functions like, for example, the function that defines the length of a vector -- f( *x* ) = sqrt(x1^2+x2^2+x3^2) -- which does take in a vector value argument and puts out a scalar value number.
i mean, electric potential for example is a scalar function with a vector argument so i don't understand what you mean when you say it doesn't exist or there is no need for it
What you teach on this youtube channel is almost like a conspiracy in the sense that it is all cohesive but to convince other that im not a crazy person id have to teach them all you have taught me (not an easy feat)
How can you say that you don't need functions with vector arguments? All of linear algebra relies on such functions.
I think he means vector arguments that don't depend further on a scalar value. Remember that we are doing differential geometry. I suspect he would be fine with a function like a reflection that has a vector argument that depends on time (or some other parameter). Those are the linear transformations from linear algebra, and you can certainly take a derivative with respect to whatever parameter you have used.
Man, don't even try to understand. This is a weird place.
Excellent point and I should make another video that clarifies this point. In this course, I use the term "vector" in the narrow sense of directed segment. It is explained in this video: czcams.com/video/N32KI6qoeRA/video.html.
Even with that definition, you could conceive functions of vector argument - for example "length" is a function of a vector. So when I say that "there is no such thing as a function of a vector argument", I mean "for the purposes of this narrative, there is no such thing as a function of a vector argument". And that's good news - fewer types of objects makes for a simpler framework.
Amazing to see these new videos, thanks so much for sharing them, always a delight to watch you bring this wonderful topic to life.
Thank you, much appreciated!
Thanks!
Of course you need f:R^n->R^m, and of course you can take its directional derivative (supposing it’s differentiable). f’(x_1, …,f_n) in the y (vector in R^n) direction will be a vector in R^m whose i-th component is the dot product . The total derivative will be a linear operator. The subject is comprehensively treated in Apostol’s volume 2.
You're correct. Please see the pinned comment.
5:32 I assume you're not talking about functions like, for example, the function that defines the length of a vector -- f( *x* ) = sqrt(x1^2+x2^2+x3^2) -- which does take in a vector value argument and puts out a scalar value number.
Good point! See my reply to the pinned comment.
Wouldnt you be able to treat the division as an inversion, like a matrix inverse?
I'm always open to ideas! Go ahead an propose an operation!
i mean, electric potential for example is a scalar function with a vector argument so i don't understand what you mean when you say it doesn't exist or there is no need for it
You're correct. Please see the pinned comment.
👍
What you teach on this youtube channel is almost like a conspiracy in the sense that it is all cohesive but to convince other that im not a crazy person id have to teach them all you have taught me (not an easy feat)
No.
It's just a perspective. In other courses, I describe different perspectives.
Uh, are you not familiar with fields? We use a gradient instead of a derivative. Functions that don't have derivatives exist.
See my reply to the pinned comment. A later video will deal with the gradient.