Recent content by ianchenmu

Homework Statement The question is: Let \phi: \mathbb{R}^n\rightarrow\mathbb{R}^n be a C^1 map and let y=\phi(x) be the change of variables. Show that dy_1\wedge...\wedge dy_n=(detD\phi(x))\cdotdx_1\wedge...\wedgedx_n. Homework Equations n/a The Attempt at a Solution Take a look at here and...

Homework Statement The question is: Let C be a symmetric matrix of rank one. Prove that C must have the form C=aww^T, where a is a scalar and w is a vector of norm one. Homework Equations n/a The Attempt at a Solution I think we can easily prove that if C has the form...

Homework Statement The question is: Suppose that lim x_k=x_*, where x_* is a local minimizer of the nonlinear function f. Assume that \triangledown^2 f(x_*) is symmetric positive definite. Prove that the sequence \left \{ f(x_k)-f(x_*) \right \} converges linearly if and only if \left...

Homework Statement 1) Suppose that f_k is integrable on [a_k,\;b_k] for k=1,...,n and set R=[a_1,\;b_1]\times...\times[a_n,\;b_n] . Prove that \int_{R}f_1(x_1)...f_n(x_n)d(x_1,...x_n)=(\int_{a_1}^{b_1}f_1(x_1)dx_1)...(\int_{a_n}^{b_n}f_n(x_n)dx_n) 2)Compute the value of the improper...

Homework Statement The question is: Let ##\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}##. Prove that if ##E\subset\pi## is a closed Jordan domain, and ##f:E\rightarrow\mathbb{R}## is Riemann integrable, then ##\int_{E}f(x)dV=0##. Homework Equations n/a...

Homework Statement |\;| is a norm on \mathbb{R}^n. Define the co-norm of the linear transformation T : \mathbb{R}^n\rightarrow\mathbb{R}^n to be m(T)=inf\left \{ |T(x)| \;\;\;\; s.t.\;|x|=1 \right \} Prove that if T is invertible with inverse S then m(T)=\frac{1}{||S||}. Homework...

Homework Statement Let G\subset L(\mathbb{R}^n;\mathbb{R}^n) be the subset of invertible linear transformations. a) For H\in L(\mathbb{R}^n;\mathbb{R}^n), prove that if ||H||<1, then the partial sum L_n=\sum_{k=0}^{n}H^k converges to a limit L and ||L||\leq\frac{1}{1-||H||}. b) If A\in...