Recent content by vintwc

ok so linear bijection is an isomorphism. i define f(v_1,v_2)=(u(v_1,v_2),0) but I'm still not sure how to proceed from there.

Homework Statement Let V be a vector space over K. Let L(V) be the set of all linear maps V->V. Prove that L(V) is a ring under the operations: f+g:x -> f(x)+g(x) and fg:x -> f(g(x)) Now, let V=U+W be the direct sum of two vector spaces over K such that the dimension of both U and W are...

Homework Statement If f:\mathbb{R} \to \mathbb{R} is such that for all r>0 there exists a continuous function g_r \mathbb{R} \to \mathbb{R} such that |g_r (x) - f(x)| < r for |x| < 1 then f is continuous at 0. Homework EquationsThe Attempt at a Solution When |x| < \delta _g, |g_r (x) - g_r...

I got that (mathmathmad's) as well but I chose x such that I could evaluate without using a calculator. It's not too difficult to be honest. What concerns me is that do we need to prove that sin(x) and its derivatives of all orders are continuous in the given domain or should we take it as given.

Ah, too late for correction. But I had the same approach as yours. I was considering cylindrical coordinates but I was having trouble with finding the limits (spherical one as well).

I got a feeling its z+R. Could anyone let me know what are the phi limits of integration?

Alright following the my notes so far which had a sort of similar but different question, I guess that the height is z+R? Assuming if this is right, the limits of integration will be [0,R]x[-Pi/2, 0]x[0,2*Pi] (since we are looking at the lower hemisphere). However, if I try to calculate the...

Homework Statement Let Ω be a tank whose shape is that of the lower hemisphere of radius R. The tank with a muddy suspension whose density ρ is ρ(x,y,z):=e^-h(x,y,z), where h(x,y,z) is the height of (x,y,z) above the lowest point of the tank. Find the center of mass in the tank Homework...

Homework Statement Let Ω be a tank whose shape is that of the lower hemisphere of radius R. The tank with a muddy suspension whose density ρ is ρ(x,y,z):=e^-h(x,y,z), where h(x,y,z) is the height of (x,y,z) above the lowest point of the tank. Find the center of mass in the tankHomework...

differentiating is a much better option fyi

Yes it is

I do not think so as it is not related with F(x) being maximum of g(a). Btw blackscorpian, I've sent you a pm.

How did you get 0.052? Mean=0, SD= sqrt(42)?

You are right for the limits of x but I think the y limits should be the way I first wrote it? Did you calculate the gradient wrongly? Btw, thanks for your help so far.

Right, so should my integral look like as follows? ∫∫ e^x dxdy with x=4-2y to y+1, y=0 to 1 + ∫∫ e^x dxdy with x=1-2y to y+4, y=-1 to 0