The jacobian matrix of partial derivatives?

[andlook]

In differential geometry what does df mean as in


<br /> <br /> \mbox{f} : \mathbb{R}^m \mbox{ to } \mathbb{R}^n

Then df is what? the jacobian matrix of partial derivatives?

 

[quasar987]

More generally, suppose the map f is a differentiable map is defined on an open set U of R^m:
f:U\rightarrow\mathbb{R}^n

In differential geometry, we define on each point x of U the tangent space to U at x as the vector space T_x\mathbb{R}^m consisting of pairs (x,v), where v\in\mathbb{R}^m and where addition and scalar multiplication are defined by (x,v) + (x,w) = (x,v+w) and a(x,v) = (x,av). A vector (x,v)\in T_x\mathbb{R}^m is to be interpreted as "the vector v "at" x". Then we form the tangent bundle of U
<br /> TU:=\bigcup_{x\in U}T_xU<br />
Then df is defined as the map df:TU\rightarrow T\mathbb{R}^n that associates to (x,v)\in T_xU the element (f(x),Df(x)v)\in T_{f(x)}\mathbb{R}^n, where Df(x) is the usual derivative of f at x in the sense of calculus or analysis.

 

[HallsofIvy]

In differential geometry what does df mean as in


<br /> <br /> \mbox{f} : \mathbb{R}^m \mbox{ to } \mathbb{R}^n

Then df is what? the jacobian matrix of partial derivatives?

Strictly speaking, df is the linear transformation that best approximates f (in the same way that y= mx+ b best approximates f(x) at x= a when m= f'(a)). Given the standard bases for R^m and Rⁿ, that is represented by the Jacobian matrix.