## Hotovy’s differentials/test

I’m going to test to see if I can post some of my write up from the chat I gave to you guys on differentials and how difficult it is to get the TeX code to transfer properly…

We should want our tangent subspace at $a$ to consist of straight lines through $f(a)$ that are tangent to the image surface $S$ under a function $f$ near $f(a)$. Well, let’s consider different curves on this ”surface”! Define

$\gamma_v(t):=f(\mathbf{a}+t\mathbf{v})$

so $\gamma$ is simply the image of a straight line through $\mathbf{a}$ under $f$. We’ll define the directional derivative with respect to $v$ of $F$ at $a$ to be the vector

$D_vf(\mathbf{a})=\lim_{h\to0}\frac{f(\mathbf{a}+h\mathbf{v})-f(\mathbf{a})}{h}$

In other words, this is the tangent vector to the image of a straight line though the point $a$. Intuitively, we want our tangent subspace to contain, in some sense, all of these directional derivative vectors, meaning if we take any line through $a$, the tangent vector to the image of that line should be contained in our tangent subspace.

Takes a bit of work to make all this work, especially centering individual lines. I had to go into the HTML editor and use HTML tags. Also the $\LaTeX$ looks kind of funny (lower than everything else), but overall not terrible. Biggest annoyance is having to go in and add the “latex” to every dollar signed thing in your latex code.

Hotovy: You can do a find/replace to get your $\TeX$ code to be WordPress-appropriate. Surely someone has written up how to do this and posted it on the web somewhere.