Fdtxjr

Can I rely on these GitHub repository files?

Do all polymers contain either carbon or silicon?

What is the term when two people sing in harmony, but they aren't singing the same notes?

Lightning Web Component - do I need to track changes for every single input field in a form

Books on the History of math research at European universities

Why are on-board computers allowed to change controls without notifying the pilots?

Installing PowerShell on 32-bit Kali OS fails

Golf game boilerplate

How to prevent YouTube from showing already watched videos?

Can the electrostatic force be infinite in magnitude?

Can somebody explain Brexit in a few child-proof sentences?

How do ultrasonic sensors differentiate between transmitted and received signals?

Is there an Impartial Brexit Deal comparison site?

For airliners, what prevents wing strikes on landing in bad weather?

"lassen" in meaning "sich fassen"

Have I saved too much for retirement so far?

Adding empty element to declared container without declaring type of element

Why is delta-v is the most useful quantity for planning space travel?

Can a Bard use an arcane focus?

Is a naturally all "male" species possible?

Invariance of results when scaling explanatory variables in logistic regression, is there a proof?

Can I Retrieve Email Addresses from BCC?

Greatest common substring

How to check participants in at events?

Parsing notation for covariant derivative embedded in ambient euclidean space

Covariant derivative (or connection) of and along a curveBetter proof that vector fields on submanifolds extend globally iff submanifold is closedShow that the outward unit normal is smooth vector fieldCovariant derivative from an Ehresmann connection on a fibre bundleCovariant derivative of contracted tensor productCovariant derivative derived from embedding in higher-dimensional euclidean spaceCovariant derivative in Euclidean spaceComputing pullbackIntrinsic definition of Jacobian matrix on manifoldsCovariant Derivatives unaffected by change in basis

0

$begingroup$

$(triangledown_XY)_p=fracDdt|_p Y=pi(fracddt|_t=0(Y(gamma(t))=pi(dY_p(gamma'(0))$

Where $x_p = gamma'(0)$

So i'm still being thrown off sometimes by seeing vector fields as functions in $mathbbR^n$ and other times using them as derivative operators. My main question here is how exactly am I to understand $dY_p$?

Like this?:

$Y(p) = (y_1(p),..,y_n(p)) rightarrow dY_p=(dy_1(p),....,dy_n(p))$

so just taking the derivative of the components of $Y$?

And then from here, would we have $dY_p$ act on $gamma'(0)=x_p?$ I guess the obvious answer is just to write $dY_p$ as a jacobian matrix and $x_p$ as a vector and use matrix multiplication.

Is this all correct thinking?

Is this all there is to taking derivatives along vector fields when embedded in euclidean space? It's just one vector field you take the total derivative normally, and then apply that total derivative into whatever direction your other vector field at that point is pointing at and project to the tangent plane? Thanks

differential-geometry

share|cite|improve this question

asked Mar 16 at 23:06

Mathematical MushroomMathematical Mushroom

918

$endgroup$

add a comment | 

0

$begingroup$

$(triangledown_XY)_p=fracDdt|_p Y=pi(fracddt|_t=0(Y(gamma(t))=pi(dY_p(gamma'(0))$

Where $x_p = gamma'(0)$

So i'm still being thrown off sometimes by seeing vector fields as functions in $mathbbR^n$ and other times using them as derivative operators. My main question here is how exactly am I to understand $dY_p$?

Like this?:

$Y(p) = (y_1(p),..,y_n(p)) rightarrow dY_p=(dy_1(p),....,dy_n(p))$

so just taking the derivative of the components of $Y$?

And then from here, would we have $dY_p$ act on $gamma'(0)=x_p?$ I guess the obvious answer is just to write $dY_p$ as a jacobian matrix and $x_p$ as a vector and use matrix multiplication.

Is this all correct thinking?

Is this all there is to taking derivatives along vector fields when embedded in euclidean space? It's just one vector field you take the total derivative normally, and then apply that total derivative into whatever direction your other vector field at that point is pointing at and project to the tangent plane? Thanks

differential-geometry

share|cite|improve this question

asked Mar 16 at 23:06

Mathematical MushroomMathematical Mushroom

918

$endgroup$

add a comment | 

$begingroup$

$(triangledown_XY)_p=fracDdt|_p Y=pi(fracddt|_t=0(Y(gamma(t))=pi(dY_p(gamma'(0))$

Where $x_p = gamma'(0)$

So i'm still being thrown off sometimes by seeing vector fields as functions in $mathbbR^n$ and other times using them as derivative operators. My main question here is how exactly am I to understand $dY_p$?

Like this?:

$Y(p) = (y_1(p),..,y_n(p)) rightarrow dY_p=(dy_1(p),....,dy_n(p))$

so just taking the derivative of the components of $Y$?

And then from here, would we have $dY_p$ act on $gamma'(0)=x_p?$ I guess the obvious answer is just to write $dY_p$ as a jacobian matrix and $x_p$ as a vector and use matrix multiplication.

Is this all correct thinking?

Is this all there is to taking derivatives along vector fields when embedded in euclidean space? It's just one vector field you take the total derivative normally, and then apply that total derivative into whatever direction your other vector field at that point is pointing at and project to the tangent plane? Thanks

differential-geometry

share|cite|improve this question

asked Mar 16 at 23:06

Mathematical MushroomMathematical Mushroom

918

$endgroup$

share|cite|improve this question

asked Mar 16 at 23:06

Mathematical MushroomMathematical Mushroom

918

share|cite|improve this question

asked Mar 16 at 23:06

Mathematical MushroomMathematical Mushroom

918

asked Mar 16 at 23:06

Mathematical MushroomMathematical Mushroom

918

asked Mar 16 at 23:06

Mathematical MushroomMathematical Mushroom

918