Recent content by lep11

Homework Statement Let ##f(x,y)=(xy,y)## and ##\gamma:[0,2\pi]\rightarrowℝ^2##,##\gamma(t)=(r\cos(t),r\sin(t))##, ##r>0##. Calculate ##\int_\gamma{f{\cdot}d\gamma}##. Homework EquationsThe Attempt at a Solution The answer is 0. Here's my work. However, this method requires that you are...

Could someone please verify if I got it right?

I think I got the right intuition, but I am not sure if this counterexample is rigorious enough. If we consider the function ##F##, we notice that ##F(\epsilon,-\epsilon,0)=(0,0)## ##∀\epsilon>0##. Because we can choose ##\epsilon## to be arbitrarily small, ##\nexists{R>0}## such that n-hood...

O.k. How would one proceed in this case? Should I construct a counterexample?

Homework Statement Determine if the following set of equations has unique solution of the form ##g(z)=(x,y)## in the n-hood of the origin. $$\begin{cases} xyz+\sin(xyz)=0 \\ x+y+z=0 \end{cases}$$ Homework Equations I assume I am supposed to use the implicit function theorem...

Using that lead to situation where the ##\alpha##'s weren't linearly independent.

Homework Statement Reduce ##xy+zy## to diagonal form. Homework Equations The desired diagonal form is ##Q(\vec{x})=(\alpha_1(\vec{x}))^2+...+(\alpha_k(\vec{x}))^2-(\alpha_{k+1}(\vec{x}))^2-...-(\alpha_{k+l}(\vec{x}))^2,## where ##\alpha_i## are linearly independent linear functions. Also known...

##f(0,0,\frac{1}{n},0)=\frac{1}{n^3}>0## and ##f(0,0,-\frac{1}{n},0)=-\frac{1}{n^3}<0## for all naturals ##n##. Thus ##a## must be saddle point. Correct?

Is there something unclear or is the question badly-worded? We are interested in the max/min values of the given function. Since the gradient is zero at (0,0,0,0), something could be happening at that point (local maxima,minima, saddle point). However, the hessian eigenvalues test is...

Homework Statement The task is to find the extreme values (and their nature) of the polynomial function . $$f(\vec{x})=x_1x_2+x_1^2+x_2^2+x_3^3+x_4^4.$$ The Attempt at a Solution The critical point is ##a=(0,0,0,0)##, which is the solution to ##\nabla{f(a)}=0.## If we form the Hessian matrix...

Ah, why not use the factorization instead? That method seemed a lot easier than struggling with the chain rule.

Yes. ##F(tb)=t^{\alpha}F(b)##, but how to use that? Is ##||tb||^l=|t|^l||b||##? $$\lim_{t\rightarrow0}\frac{F(tb)+G(tb)}{\|tb\|^l}=\lim_{t\rightarrow0}\frac{t^lF(b)+t{^\alpha}G(b)}{|t|^l\|b\|^l}=\lim_{t\rightarrow0}(\frac{t^lF(b)}{|t|^l\|b\|^l}+\frac{t{^\alpha}G(b)}{|t|^l\|b\|^l})=...?$$, where...

Homework Statement The task is to prove that $$\lim_{x\rightarrow0}\frac{Q_1(x)-Q_2(x)}{\|x\|^k}=0 \implies Q_1=Q_2,$$ where ##Q_1,Q_2## are polynomials of degree ##k## in ##\mathbb{R}^n##. Homework Equations $$ \lim_{x\to 0} \frac{a x^\alpha}{\|x\|^n}=\left\{\begin{array}{c} 0 \textrm{ if }...

This is hopeless. ##\frac{\partial{h}}{\partial{u}}(x,y)=\frac{\partial{f}}{\partial{x}}(x,y)+\frac{\partial{f}}{\partial{y}}(x,y)## and ##\frac{\partial{h}}{\partial{v}}(x,y)=\frac{\partial{f}}{\partial{x}}(x,y)-\frac{\partial{f}}{\partial{y}}(x,y)##...

So now ##h_u=f_1(u+v,u-v)-f_2(u+v,u-v)## and ##h_{uv}=f_{12}(u+v,u-v)(1)-f_{22}(u+v,u-v)(-1)##?