The jacobian matrix of partial derivatives?

[andlook]

In differential geometry what does df mean as in


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

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
$$TU:=\bigcup_{x\in U}T_xU$$
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


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

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.