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
Let ##E\subset\mathbb{R}^n## be a closed Jordan domain and ##f:E\rightarrow\mathbb{R}## a bounded function. We adopt the convention that ##f## is extended to ##\mathbb{R}^n\setminus E## by ##0##.
Let ##\jmath## be a finite set of Jordan domains in ##\mathbb{R}^n## that...
Homework Statement
Let T:\mathbb{R}^n\rightarrow\mathbb{R}^n be a linear transformation and R\in \mathbb{R}^n be a rectangle.
Prove:
(1) Let e_1,...,e_n be the standard basis vectors of \mathbb{R}^n (i.e. the columns of the identity matrix). A permutation matrix A is 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 p(\lambda )=\lambda^3+a_2\lambda^2+a_1\lambda+a_0=(\lambda-x_1)(\lambda-x_2)(\lambda-x_3) be a cubic polynomial in 1 variable \lambda. Use the inverse function theorem to estimate the change in the roots 0<x_1<x_2<x_3 if a=(a_2,a_1,a_0)=(-6,11,-6) and a changes by \Delta...
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...