\(\def\Real{\mathbb{R}}\)

#### Course outline is here.

#### Taylor formula

* Spaces of differentiable functions* on an interval \(I=[a,b]\subset\Real\):

\[

C(I,\Real), C^1(I,\Real),\ldots, C^n(I,\Real),\ldots

\]

For a function in \(C^1(I,\Real)\),

\[

f(y)=f(x)+\int_x^yf'(s)ds.

\]

Iterating (for functions in \(C^n(I,\Real)\), we obtain

\[

f(y)=f(x)+f'(x)(y-x)+\frac{f”(x)}{2!}(y-x)^2+\ldots+\frac{f^{(n-1)}(x)}{(n-1)!}(y-x)^{n-1}+

\int_{x<s_1<s_2<\ldots<s_n<y}f^{(n)}(s_1)\frac{(y-s_1)^{n-1}}{(n-1)!}ds.

\]

* Implications*:

- Mean value theorem
- If \(f”\geq 0\) on \(I\) and \(f'(x)=0\), then \(x\) is a global minimum of \(f\).

#### Several variables

Domain: \(U\subset\Real^n\), an open set.

* Reminder*: open, closed, bounded, compact sets.

* Exercise*: is the set \(\{|x|\leq 1, |y| \lt 1\}\) open? closed? bounded?

* Partial and directional derivatives*; \(\nabla f\).

Remark: the definition of the directional derivative depends on the other coordinates held constant.

**Gateaux and Frechet derivatives**

Gateaux derivative is in essence directional one. Frechet derivative asks for the function locally to be close, up to higher order terms;

\[

f(x+v)=f(x)+(l,v)+o(|v|).

\]

**Reminder**: \(o,O,\sim\) notation.

Theorem: Frechet differentiability implies Gateaux differentibility.

Not vice versa:

**Example**: \[f(x,y)=\frac{x^3y}{x^4+y^2}\]

Multivariate mean value:

If \(f\) is Gateaux differentiable,

\[

f(x+v)=f(x)+(f'(z),v)

\]

for some \(z\in(x,x+v)\)…

Gap between Gateaux and Frechet differentiabilit**y: **

Theorem: if Gateaux differentiable function has continuous gradient, then it is Frechet differentiable.

#### Vector-Valued Functions

Differentibility for

\[

F:\Real^n\to\Real^m

\]

carries over from scalar (\(m=1\)) case: component-wise. Derivative now is a linear function \(\Real^n\to\Real^m\); given (if we fix coordinates) as matrix.

*Jacobian* \(DF(x)\), an \(m\times n\) matrix.

*Chain rule* for (Frechet) differentiable functions:

If

\[

H=G\circ F (F:\Real^n\to\Real^m; G:\Real^m\to\Real^l; H:\Real^n\to\Real^l)

\]

then

\[

DH=DG\circ DF.

\]

**Remark**: Gateaux differentiability is not enough.

Mean value theorem also fails (example?). But one has inequalities:

\[

\parallel F(x+v)-F(x)- DF(x) v\parallel\leq L \parallel v\parallel

\]

where \(L\) is a bound on the operator norm od \(DF(z)-DF(x)\) over the segment \((x,x+v)\).

**Higher derivatives.**

Iterating, one obtains higher derivatives of multivariate functions. They are essentially symmetric polylinear forms.

They are needed to formulate a multidimensional Taylor formula.

*Taylor approximation*: polynomial approximating the function to higher order,

\[

f(x+v)=f(x)+\sum_{|k|\leq K} \frac{1}{k_1!\cdots k_n!}\prod \frac{\partial^k f}{\partial^{k_1}x_1\cdots\partial^{k_n}x_n} v_1^{k_1}\cdots v_n^{k_n}+o(|v|^K).

\]

**Examples**:

- Find coefficient \(a_{2,0,1}\) of \(\sin(2x_1-3x_2+x_3)\)
- Find gradient of \(\log\det(X)\), where \(X\) is an invertible \(n\times n\) matrix
- Find Jacobian of \(X^{-1}\).

**Exercises:**

- Find derivative of

\[

t\mapsto \exp(tA)B\exp(tC)

\]

at \(t=0\) (here \(A,B,C\) are \(n\times n\) matrices). - Find Jacobian of

\[

\left(\begin{array}{c}x\\y\end{array}\right)\mapsto\left(\begin{array}{c}x^3-3xy^2\\3x^2y-y^3\end{array}\right)

\]

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