1
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0answers
26 views

Big-O Notation for remainder terms in Taylor expansion

The Big-O notation is commonly used in Taylor expansions of the form $$f(x+\epsilon)=f(x)+\epsilon f'(x)+O(\epsilon^2)$$ to say that the remainder term grows at least quadratic around $\epsilon=0$. ...
1
vote
1answer
31 views

Partial derivatives + Taylor's Formula in several variables

Given a function $f(x) = (x_1+...+x_n)^k$, how do we show that $$D_1^{j_1}\cdots D_n^{j_n}f(x) = k!$$ if $j_1+...+j_n = k$?
1
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2answers
29 views

Third degree Taylor polynomial in two variables

How does one find the third-degree Taylor polynomial of $f(x,y) = (x+y)^3$ at the points $(0,0)$ and $(1,1)$? Many thanks
2
votes
2answers
43 views

Application of Taylor's Formula

If we are given that $f''(x) = f(x)$, how do we show that there exist constants $a$ and $b$ such that $f(x) = ae^x + be^{-x}$ for all $x$? A hint is given: We can define another function $g$ by $g(x) ...
2
votes
0answers
36 views

What is the 2nd order taylor polynomial of f(x,y)?

I'm just computing the 2nd order taylor polynomial for $f(x,y) = tan(x + 3y + \frac{\pi}{4})$ centered at (3,-1) and wondering if I have done this correctly or if anyone has any suggestions on how I ...
3
votes
1answer
33 views

Finding $\frac{\partial ^8 f}{\partial x^4\partial y^4}$

Given the function $f(x,y)=\frac{1}{1-xy}$ find the value of$\frac{\partial ^8 f}{\partial x^4\partial y^4}(0,0)$. First I developed the function into a taylor series using geometric series ...
1
vote
0answers
32 views

Multivariate Taylor Polynomial

The Exercise: Calculate the Taylor polynomial of degree 3 of $f(x,y,z)=x^5y^4z^3$ at $(1,1,1)$ in an arbitrary direction $h$. Use Taylor's theorem to get a bound on the remainder when using this ...
3
votes
2answers
123 views

On the hessian matrix and relative minima

I'm asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
0
votes
1answer
181 views

Taylor expansion of a vector field (notation question)

Is there an index-less notation (using gradiends, Jacobians, curls, hessians, anything) to describe a second-order term in the Taylor expansion of a vector field $\mathbf{f}(\mathbf{x}): \mathbb{R}^n ...
1
vote
1answer
55 views

Lagrange Taylor remainder: can we choose $t^*$ continuously?

The Taylor theorem with Lagrange remainder tells us that for $f: \mathbb{R}^n \to \mathbb{R}$ twice differentiable (we can assume $C^2$ if we like), $$f(y) - f(x) = \left\langle \nabla f(x), y-x ...
0
votes
2answers
48 views

Version of Taylor: $F(x+h)-F(x) = \left \langle \int_0^1 \nabla F(x+th)dt, h \right \rangle.$

My teacher claimed without proof that Taylor's theorem with remainder implied that for a suitable function $F: \mathbb{R}^n \to \mathbb{R}$, $$F(x+h)-F(x) = \left \langle \int_0^1 \nabla F(x+th)dt, h ...
1
vote
4answers
57 views

Differentiating second order term of Taylor polynomial (multivariable)

I am trying to derive Newton step in an iterative optimization. I know the step is: $$\Delta x=-H^{-1}g$$ where H is Hessian and $g$ is gradient of a vector function $f(x)$ at $x$. I also know the ...
4
votes
2answers
103 views

Taylor polynomial about the origin

Find the 3rd degree Taylor polynomial about the origin of $$f(x,y)=\sin (x)\ln(1+y)$$ So I used this formula to calculate it ...
0
votes
2answers
72 views

First order multivariate approximation

To demonstrate that $\nabla\!_{\hat{\boldsymbol u}}\,f(\boldsymbol{x}) \equiv \left \langle \hat{\boldsymbol u}, \nabla f(\boldsymbol{x}) \right \rangle$ I plug a first order expansion of ...
1
vote
1answer
32 views

How Hessian feature detector works?

I know about Harris corner detector, and I understand the basic idea of its second moment matrix, $$M = \left[ \begin{array}{cc} I_x^2 & I_xI_y \\ I_xI_y & I_y^2 \end{array} \right]$$, edges ...
3
votes
0answers
66 views

Taylor Polynomial in Multivariable Case

I am doing Problem 9.30 in Rudin's Principles of Mathematical Analysis and have done part (a) and (b) but got stuck on part(c). In part (b) I have finished the proof that: $$f(\mathbf a+\mathbf ...
0
votes
0answers
608 views

Taylor's Theorem for Multivariate Functions

Please look at this theorem in Wiki regarding Taylor's theorem generalized to multivariate functions: Multivariate version of Taylor's Theorem The version stated there is one that I'm not familiar ...
0
votes
1answer
67 views

Taylor expansion in $4D$

Let $f(x)=(x_2,-x_1,\sqrt 2 x_4 + x_1^3,-\sqrt 2x_3+x_3x_4^2)$ be a vector valued function from $\mathbb R^4\to\mathbb R^4$. Would anyone help me expand it up to and including the third term in its ...
0
votes
0answers
140 views

Lagrange remainder for multivariate Taylor expansion?

Assume that $f:R^n\to R$ is an analytic function and we want to use the 2nd order Taylor expansion around, let say 0. Is it correct to write the remainder in the Lagrange form? I.e. $ f(X) = f(0) + ...
1
vote
1answer
227 views

Determine whether a multi-variable limit exists

I need to determine whether the next limit exists: $$\lim_{(x,y)\to(0,0)}f(x,y)=\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}$$ Looking at the numerator $(-1-\frac{x^2}2)$ it immediately ...
2
votes
0answers
101 views

Taylor series of a vector field?

Can someone give me the definition of the Taylor Series of a vector field in $\mathbb{C}^2$? Thanks!
2
votes
4answers
661 views

What exactly are the “higher order terms” (H.O.T.) in Taylor series expansion in multivariable case?

I know the Taylor series expansion in single variable case: $$ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{3!}f^{(3)}(x_0)(x-x_0)^3 + \frac{1}{4!}f^{(4)}(x_0)(x-x_0)^4 ...
2
votes
1answer
56 views

Bounding a continuously differentiable function using Taylor given the function is bounded by the norm of x

Suppose $0 < r < 1$ and that $f \colon B_1(0) \to \mathbb R$ is continuously differentiable. If there is an $\alpha > 0$ s.t. $|f(x)<\Vert x\Vert^\alpha$ for all $x \in B_r(0)$, prove ...
1
vote
1answer
91 views

Exact expansion of functions

Prove that for any twice differentiable function $f: {R}^n \to R$, $f(y) = f(x) + \nabla f(x)^T (y-x)+ \frac{1}{2} (y-x)^T \nabla^2f(z)(y-x) $, for some $z$ on the line segment $[x, y]$. Note that ...
7
votes
1answer
228 views

taylor's formula in $\mathbb{R}^n$ as exponential of derivative operator

I hope this question is not to vague. There is some relationship between the formal operator $$e^\frac{d}{dx} = 1 + \frac{d}{dx} + \frac{1}{2!}\frac{d^2}{dx^2} + \ldots $$ and taylor's formula. Is ...
4
votes
2answers
115 views

$f(y) \leq f(x)+\nabla f(x)\cdot (y-x) $ and $f(x)\geq 0$ implies that $f$ is constant.

Here is the question. Suppose that $f: \mathbb R^n \rightarrow \mathbb R$ has two derivatives and the associated hessian matrix is negative semidefinite on all of $\mathbb R^n$. Show that for any ...
0
votes
1answer
154 views

Applying a multidimensional variant of Taylor's Theorem

Given $f:R^n\rightarrow R^n$ continuously differentiable in some convex open set $D$ and $x,x+p\in D$, taylor's theorem is given as: $$f(x+p)=f(x)+\int_0^1 J(x+tp)p \, dt,\text{ where }J\text{ is ...
1
vote
2answers
2k views

Vector taylor series

Classical Electrodynamics by Jackson says "With a Taylor series expansion of the well-behaved $\rho (\mathbf{x'})$ around $\mathbf{x'} = \mathbf{x}$ one finds ..." and then he says basically that ...
2
votes
1answer
78 views

When is the limit in $y$ of a Taylor expansion in $x$ a valid expansion?

I'd be interested to know when, if $$f(x,y)=g(x,y)+O(x^n)$$ we have that $$\lim_{y\rightarrow c}=\lim_{y\rightarrow c}g(x,y)+O(x^n).$$ Are there conditions of $f$ and/or $g$ that make sure that this ...
1
vote
2answers
101 views

Taylor's formula

Taylor's Formula Write taylor's formula for $F(x,y)= \sin(x)\sin(y)$ using $a=0$, $b=0$, and $n=2$. $$\sin(h)\sin(k)=hk−\frac 16h(h^2+3k^2)\cos\theta h\sin\theta k−\frac 16 k(3h^2+k^2)\sin\theta ...
1
vote
1answer
817 views

how do I find the taylor polynomial of multivariable functions?

I know taylor polynomial for single variable functions but I am having trouble understanding how to find taylor polynomials for multivariable functions. I know how to find partial derivatives as well ...
1
vote
1answer
124 views

Product of Taylor polynomials

I'm trying to prove the following proposition: Let $U\in R^n$ be open, and $f,g\colon U\to R$ be $C^k$ functions, then the Taylor polynomial of $fg$ is computed as $P_{f,a}^k(a+\vec{h})\cdot ...
1
vote
1answer
227 views

Taylor series expansion of $\sec(x +y^2)$

We have $f(x,y) = \sec(x+y^2)$ I want to find the first two non-zero terms of $f$ at $(0,0)$ starting by Taking the first few terms of $\cos x$ centered at zero, $1 - \frac{x^2}{2!} $ Using this ...
0
votes
2answers
150 views

Domain of convergence of $f^{-1}: \mathbb R ^N \mapsto \mathbb R^N$ taylor series

In another question, I ask about the topology of the singular manifold of the Jacobian. What i want to ask in here is about the radius of convergence of a Taylor series expansion of the inverse ...
1
vote
1answer
228 views

Taylor expansion in time of the time component of a stress energy tensor

Perform a taylor expansion in 3 dimensions in time on the time compontent of of $T^{\alpha \beta}(t - r + n^{i} y_{i})$ given that $r$ is a contstant and $n^{i} y_{i}$ is the scalar product of a ...