Differential Geometry: Lecture 2 part 1: points, vectors, directional derivative

Here I introduce the notation for points, tangent vectors, tangent space, the tangent bundle and vector fields. Some general comments about orthogonal complements in the tangent space at p are offered. Part 2 continues this.

[the_informative_edge]

The best explanation for the basic notations used in Differential Geometry... Thank you James

[henry]

Thank you so much for this helpful series of videos.
Besides the math aspects I love this video at 8:02 when the Mr. Ant comes in.

[RalphDratman]

I'm glad you deal with the difference between an ordinary origin-rooted vector and a tangent space vector. Those roaming vectors used to bother me in physics, because clearly what they called a vector was not the same as the concept of an element of a vector space in math.

[SeBeom]

With your video, I am getting a great help in my school class. I learned linear algebra and calculus concepts in university, but I had difficulty learning because I couldn't learn enough Your lectures have helped me to improve my grades.

[collegemathematics]

thak you james your lectures encouraged me to study this buteful topic

[cihantemel]

i have two problems in this video,
1) when we mention about per sets for curve in R^3, why do we take a flat area as (s1)
2) i don't comprehend what's the relation between tanjant space and direct sum

[vinaysipani]

At 8:45
Shouldn't that be
TR^n = union of all p's ({p} × R^n)

[lid]

With regards to the end where you say the vector field X is a section of TS, what does that mean? What is a ''section''? thanks for the videos by the way!

[md]

How can you call something a "tangent vector" without specifying what it is tangent to? That's just seems meaningless. I bought O'Neill to try to learn this stuff myself and he does that immediately in section 1.2.... WTH??

[fernandoiglesiasg]

"Let me not distract you with technical things" :-/

[AbhishekMahajan]

You should clear your point...7.22 everything is messing up

[awesomtacular]

I have watched the beginning of several of your differential geometry classes about tangent spaces and normal spaces. In each class you seem to get lost in set theoretic definitions of spaces where you use different notations and definitions in the same classes and sometime same lectures and then have to correct yourself and have people correct you in the comments. Ive watched other classes of yours and they are great. You should rethink the way you launch into differential geometry. Sorry to be brutal.

[udit]

I don't understand how

(P+Q)^i=P^i+Q^i

[jacobvandijk]

Why didn't you prepare a little better? As soon as it is abstract you're okay, but outside of it ... :-(