Recent content by gotmilk04

We aren't given that f is continuous, which is why I'm stuck.

Homework Statement Show that if (x_{n}) is a sequence in a metric space (E,d) which converges to some x\inE, then (f(x_{n})) is a convergent sequence in the reals (for its usual metric). Homework Equations Since (x_{n}) converges to x, for all ε>0, there exists N such that for all...

So then ε_{ijkl}a^{i}a^{j}= -ε_{jikl}a^{i}a^{j}?

Homework Statement If a, b, and c are any three vector fields in locally Minkowskain 4-manifold, show that the field ε_{ijkl}a^{i}b^{k}c^{l} is orthogonal to \vec{a}, \vec{b}, and \vec{c}. Homework Equations The Attempt at a Solution I know I have to show that multiplying the...

Homework Statement Consider the ellipsoid L \subsetE3 specified by (x/a)^2 + (y/b)^2 + (z/c)^2=1 (a, b, c \neq 0). Define f: L-S^{2} by f(x, y, z) = (x/a, y/b. z/c). (a) Verify that f is invertible (by finding its inverse). (b) Use the map f, together with a smooth atlas of S^{2}, to...

I multiplied it by the complex conjugate, but there are still i's in the equation. Aren't there supposed to be no i's?

If I multiply by the complex conjugate, I'll get |\Psi(x,t)|^{2}?

Homework Statement One of the quantum mechanics wave functions of a particle of unit mass trapped in an infinite potential square well of width 1 unit is given by Ψ(x,t)= sin(\pix)e^{-i(\pi^2\overline{h}/2)t} + sin(2\pix)e^{-i(4\pi^2\overline{h}/2)t}\ where \overline{h} is a certain...

Homework Statement Define f as: f(x)= 2 if 0\leqx<1 f(1)=0 f(x)= -1 if 1<x<2 f(2)= 3 f(x)=0 if 2<x<3 f(3)=1 Prove f is integrable using six subintervals and find the value of \intf(x) dx The Attempt at...

Homework Statement Prove Q[sqrt 2, sqrt 3] is a field. Homework Equations Q[sqrt 2, sqrt 3]= {r + s\sqrt{2} + t\sqrt{3} + u\sqrt{6}| r,s,t,u\in Q} The Attempt at a Solution I know I have to show each element has an inverse, but I don't know how on these elements.

Smallest subfields containing Z[i] and Z[sqrt2] Homework Statement What are the smallest subfields of R containing Z[i] and Z[\sqrt{2}]? Homework Equations Z[i]= {a+ib|a,b\in Z} Z[\sqrt{2}]={a+b\sqrt{2}|a,b\in Z} The Attempt at a Solution Z[i]\subsetQ[i] and...

Homework Statement Give an example of a ring monomorphism f:M(2;R)-M(3;R) Homework Equations The Attempt at a Solution I can't think of anything that would be a monomorphism.

So for part a), I would do let a,b\in R and f(a),f(b)\in R' So f(a)-f(b)= f(a-b) \in R' and f(a)f(b)= f(ab)\in R'?

Homework Statement Prove that if f:R-R' is a ring homomorphism, then a) f(R) is a subring of R' b) ker f= f^{-1}(0) is a subring of R c) if R has 1 and f:R-R is a ring epimorphism, then f(1_{R})=1_{R'} Homework Equations For a ring homomorphism, f(a+b)= f(a) + f(b) f(ab)= f(a)f(b)...

Homework Statement Write the multiplication table of C_{6}/C_{3} and identify it as a familiar group. Homework Equations The Attempt at a Solution C_{6}={1,\omega,\omega^2,\omega^3,\omega^4,\omega^5} C3={1,\omega,\omega^2} The cosets are C3 and \omega^3C3 I just need help...