@@MichaelPennMath This is the definition I have: Let X be a subspace of a space E ( E can be a normed one) an element v from E is said to be a tangent vector to X in x if there existe epsilon strictly big than 0 and a parametric arc (gamma) defined from the open interval (-epsilon,+epsilon) to X differentiable in 0 such that gamma(0)=x and the derivative of gamma in 0 equal v. I did my best in the translation..
It still feels like this is a tangent vector at a point x\in X. If I am understanding: E is some "large" ambient space (like R^n or C^n) and X is a subset (probably like a smooth manifold), then v\in E is tangent to X at x if (your definition). One thing that you can do is put all of these vectors together into a tangent space -- the tangent space of X at x, sometimes denote T_x(X). This is an important concept in Lie theory -- the Lie algebra of a Lie group is the tangent space at the identity.
@@MichaelPennMath for example if I wanted to find all the tangent vector to [-1,1]^2 at (0.0) should I look for a tangent space? ( I'm still learning this concept so I'm not very comfortable with)
@@nailabenali7488 From your definition, any curve in R^2 is tangent to this space at (0,0). You can take your curve to be a line through the origin in the direction of whatever vector you want. I think this may be an example of something that is very simple, but not realistic so it makes it tricky.
I presume because he said the magnitude doesn't matter and since the x and z were equal to each other and the y would always be 0, he just set it to 1,0,1 for simplicity
Great explanation. Love from Canada
Thank you. Greetings from Iran.
so clear, thank you
This video seems to be put too early in the playlist. Chain rule as of 2:40 not proved yet, gradient not defined.
These series is really good. Can you recommend textbooks on multivariable calculus in IR^n
Hey! There is on my course a notion called tangent vectors over a subset!! It's really different from this, do you have any idea on it?
I am not really sure about this - I would have to see how it is precisely defined.
@@MichaelPennMath This is the definition I have: Let X be a subspace of a space E ( E can be a normed one) an element v from E is said to be a tangent vector to X in x if there existe epsilon strictly big than 0 and a parametric arc (gamma) defined from the open interval (-epsilon,+epsilon) to X differentiable in 0 such that gamma(0)=x and the derivative of gamma in 0 equal v. I did my best in the translation..
It still feels like this is a tangent vector at a point x\in X. If I am understanding: E is some "large" ambient space (like R^n or C^n) and X is a subset (probably like a smooth manifold), then v\in E is tangent to X at x if (your definition). One thing that you can do is put all of these vectors together into a tangent space -- the tangent space of X at x, sometimes denote T_x(X). This is an important concept in Lie theory -- the Lie algebra of a Lie group is the tangent space at the identity.
@@MichaelPennMath for example if I wanted to find all the tangent vector to [-1,1]^2 at (0.0) should I look for a tangent space? ( I'm still learning this concept so I'm not very comfortable with)
@@nailabenali7488 From your definition, any curve in R^2 is tangent to this space at (0,0). You can take your curve to be a line through the origin in the direction of whatever vector you want. I think this may be an example of something that is very simple, but not realistic so it makes it tricky.
awesome bro thanks!!
I wish you were my multivariable calc. lecturer
Great vid
how and why did you scale = , thanks by the way, great videos
I presume because he said the magnitude doesn't matter and since the x and z were equal to each other and the y would always be 0, he just set it to 1,0,1 for simplicity
so you ignore the k?