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Let $\varphi:X\to Y$ be a smooth map between finite dimensional vector spaces over $\mathbb R$. Let us define its differential in a pont $a\in X$ as a (linear) map $d\varphi(a):X\to Y$ acting by formula $$ d\varphi(a)(p)=\lim_{t\to 0}\frac{\varphi(a+tp)-\varphi(a)}{t},\qquad p\in X. $$ For each $k\in{\mathbb N}$ the map $d\varphi(a):X\to Y$ generates a map $\Lambda_kX\to \Lambda_kY$ between the spaces of multivectors of degree $k$: $$ x_1\wedge...\wedge x_k\quad \mapsto\quad d\varphi(a)(x_1\wedge...\wedge x_k)=d\varphi(a)(x_1)\wedge...\wedge d\varphi(a)(x_k) $$ Questions:

  • What is this map $$ x_1\wedge...\wedge x_k\quad\mapsto\quad d\varphi(a)(x_1\wedge...\wedge x_k) $$ called (at least for $k=\dim X$)? (Some special kind of derivative of $\varphi$ in $a$?)
  • Does it have a standard notation?
  • Can anybody advice any reading about this?
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This is called the natural extension of $d\varphi$ to the exterior powers that (at least in differential geometry) are denoted by $\Lambda^k(X)$ and $\Lambda^k(Y)$. It is linear algebra between the tangent spaces $T_pX$ and $T_{\varphi(p) }Y$. In linear algebra a linear map $f: V \to W$ between vector spaces induced linear maps (sometimes by abusing the notation also denoted by $f$) between the tensor products associated to $V$ and $W$ e.g. $ f : V \otimes V \to W \otimes W $ defined as $f(v \otimes w) := f(v) \otimes f(w)$ and then extended by linearity, see for example: https://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_linear_maps

Then you apply the above construction from linear algebra to the differential of $d\varphi : T_pX \to T_{\varphi(p)}$ to get the natural extensions.

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  • $\begingroup$ As far as I understand, this is a generalization of the notion of Jacobian determinant of $\varphi$ in $a$. Does this generalization have a name? en.wikipedia.org/wiki/Jacobian_matrix_and_determinant $\endgroup$ – Sergei Akbarov Jun 24 '15 at 10:35
  • $\begingroup$ You have the Jacobian determinant when $dim(X)=dim(Y)=n$ since the top exterior powers $\Lambda^n(X), \Lambda^n(Y)$ are 1 dimensional and a linear map between 1 dimensional vector spaces is when writen w.r.t. basis just the multiplication by a number. Such a number is the determinant you know. I think that people working in algebraic geometry call this special case of the natural extension the determinant map associated to $\varphi$. Namely, there are research areas in math in which some natural extension has a concrete name, but I don't think such names are globally accepted. $\endgroup$ – Holonomia Jun 24 '15 at 10:47

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