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Hadamard’s lemma

Hadamard’s lemma: Any smooth function $f$ in a starlike neighborhood of a point $z$ is representable in the form

$$f(x)=f(z) \sum_{i=1}^n

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(x_i-z_i)g_i(x), \;g_i\in {C^{\infty}(U)},$$
where $g_i$ are smooth functions.

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signs):

 

$$Int_Momega = 0$$

$Int_Momega = 0$

  1. $x^2 y^2=1$
  2. $f: C mapsto R$
  3. $Spec M$

The above should work after we enable simple math jax plugin and configure it with proper commands.

Some XY-commutative diagram (in double dollar signs and xymatrix {..}, begin{xy} is not necessary, but does no harm note that html direct lender payday loans tags break it, so multiline formulas should be entered via Text

tab, its much better to keep it one line):
$$\begin{xy}
\xymatrix {U ar@/_/[ddr]_y ar@{.>}[dr]|{\langle x,y \rangle} ar@/^/[drr]^x \
& X \times_Z Y ar[d]^q ar[r]_p & X ar[d]_f \& Y ar[r]^g & Z}
\end{xy}$$