Intereting Posts

What is a Manifold?
How to factor the ideal $(65537)$ in $\mathbb Z$?
When $\operatorname{Hom}_{R}(M,N)$ is finitely generated as $\mathbb Z$-module or $R$-module?
Index notation for tensors: is the spacing important?
In which topological spaces is every singleton set a zero set?
meaning of normalization
Symmetric random walk with bounds
Is there a more elegant way of proving $\langle (1,2)(3,4), (1,2,3,4,5) \rangle = A_5$
N unlabelled balls in M labeled buckets
Why do units (from physics) behave like numbers?
Derivation of the Riccati Differential Equation
Closed form for ${\large\int}_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$
Rate of convergence of modified Newton's method for multiple roots
The longest string of none consecutive repeated pattern
Let $A$ be a square complex matrix of size $n$ then $\langle Ax,x\rangle=0$ for $\forall x \in \Bbb C^n$ $\iff A=0$

I am having trouble understanding, why the $k$-th derivative of a map $F\colon\mathbb R^n \to\mathbb R^m$ is a symmetric multilinear map for each $x$ in $\mathbb R^n$. Can you please explain which vectors this map accepts as input where multilinearity comes from ? Also, why is symmetry mentioned ?

Thank you

readingframe

- Differentiation using first principles with rational powers
- Proving that an integral is differentiable
- Deriving the Normalization formula for Associated Legendre functions: Stage $3$ of $4$
- Sobolev meets Wiener
- Is there any geometric explanation of relationship between Integral and derivative?
- Partial derivatives and orthogonality with polar-coordinates

- Question regarding differentiation
- Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.
- Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
- Use the definition of the derivative to find $f'(x)$ for $f(x)=\sqrt{x-2}$
- Finding the $n$-th derivatives of $x^n \ln x$ and $\frac{\ln x}{x}$.
- How to differentiate $\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{…}}}}_{n\text{ times}}$?
- Define second derivative ($f''$) without using first derivative ($f'$)
- Question about the derivative definition
- Derivative of matrix involving trace and log
- Higher order derivatives of the binomial factor

Well, in the general case where $F:\mathbb R^n\to V$ for some (normed) vector space $V$, the derivative $F'(x_0)$ at some point $x_0\in\mathbb R^n$ is a linear map $\mathbb R^n\to^{\mathrm{Lin}} V$ such that

$$F(x_0+h)=F(x_0) + F'(x_0)(h) + o(h)$$

when $h\in\mathbb R^n$ goes toward $0$.

Thus, the function that takes every $x_0$ to the derivative at $x_0$ is of type $\mathbb R^n\to(\mathbb R^n\to^{\mathrm{Lin}} V)$. Because the space of $\mathbb R^n\to^{\mathrm{Lin}} V$ functions is itself a vector space, we can repeat the process to get higher-order derivatives. The $k$th derivative, if it exists, ends up having type

$$\mathbb R^n \to (\underbrace{\mathbb R^n\to^{\mathrm{Lin}}(\mathbb R^n\to^{\mathrm{Lin}}(\cdots\to^{\mathrm{Lin}}(\mathbb R^n\to^{\mathrm{Lin}}}_{k\text{ times}} V)\cdots)))$$

which (if you know tensor products) is isomorphic to

$$\mathbb R^n \to (\underbrace{\mathbb R^n\otimes\mathbb R^n\otimes\cdots\otimes\mathbb R^n}_{k\text{ times}}\to^{\mathrm{Lin}} V)$$

Thus, for each $x_0$, $F^{(k)}(x_0)$ is a multilinear map taking $k$ vectors in $\mathbb R^n$ to one vector in $v$.

That the map is symmetric means that

$$F^{(k)}(x_0)(h_1)(h_2)\cdots(h_k)=F^{(k)}(x_0)(h_{\sigma(1)})(h_{\sigma(2)})\cdots(h_{\sigma(k)})$$

for any permutation $\sigma$ of the $h_i$’s. (This is true under appropriate smoothness conditions for the original $F$).

**Generalizes ordinary higher derivatives.** When $n=1$, this all reduces to ordinary higher derivatives because $\mathbb R\to^{\mathrm{Lin}} V$ is naturally isomorphic to $V$ (each $f:\mathbb R\to^{\mathrm{Lin}} V$ is fully determined by $f(1)$).

**Connection to partial derivatives.** A multilinear map is determined by its values on the standard basis vectors, and

$$F^{(k)}(\vec x_0)(\mathbf e_i)(\mathbf e_j)\cdots(\mathbf e_l) =

\frac{\partial^k F}{\partial x_i \partial x_j \cdots \partial x_l} (\vec x_0) $$

- Is the set of all functions from $\mathbb{N}$ to $\{0,1\}$ countable or uncountable?
- Determinant of a generalized Pascal matrix
- Very confused about a limit.
- What is the difference between advanced calculus, vector calculus, multivariable calculus, multivariable real analysis and vector analysis?
- An Identity Concerning the Riemann Zeta Function
- Striking applications of integration by parts
- Prove that if $f$ is integrable on $$ then so is $|f|$?
- combinatorial question (sum of numbers)
- Is a semigroup $G$ with left identity and right inverses a group?
- Prove $\sum_{i=0}^{n}\left(x_{i}^{n}\prod_{0\leq k\leq n}^{k\neq i}\frac{x-x_k}{x_i-x_k}\right)=x^n$
- Factorization of a map between CW complexes
- Resolution of Singularities: Base Point
- A question on $p$-groups, and order of its commutator subgroup.
- continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$
- How do we show every linear transformation which is not bijective is the difference of bijective linear transforms?