Recent content by mathmonkey

Hi all, I'm looking to pick up a good textbook specifically focusing on the physics of energy (bonus points if it also includes some engineering applications), meant for someone at the level of first or second year physics undergrad. Ideally, I would've liked something like this course at MIT...

Thanks guys! That makes perfect sense!

Hi Mandelbroth, Thanks for your reply! However, that picture only describes the limsup/liminf of a sequence of points, which is itself a point, which is easier to intuit for me. But what I'm wondering is what is the limsup/liminf of a sequence of functions supposed to be? Is it to be a function...

Homework Statement Hi I've come across the term lim inf ##f_n## in my text but am not sure what it means. ##\lim \inf f_n = \sup _n \inf _{k \geq n} f_k## In fact, I am not sure what is supposed to be the output of lim inf f? That is, is it supposed to return a real-valued number, or a...

Aha! Thank you! My boneheaded self kept reading "step function" but thinking "simple function" :redface:

Homework Statement Let ##f## be a bounded function on [a,b] and let ##h## be the upper envelope of ##f##. Then ##R \overline{\int}_a^b f = \int _a^b h. ## (if ##\phi \geq f ## is a step function, then ##\phi \geq h## except at a finite number of points, and so ##\int _a^b h \leq R...

Ah never mind, I've figured it out. The better way is to use polar coordinates for the area of the plane. The integral simplifies to ## F = MGb\sigma \int _{0}^{2\pi} \int _{0}^{\infty} r/(r^2+b^2)^{3/2}drd\phi ## Thanks for the help.

Hmm, then maybe I am having trouble with simplifying the integral. What is the most efficient way to do it? What I thought to do is this: ##F = MG\sigma \int _A \cos (\theta) / R^2 dA ## ##F = MG\sigma \int _A R\cos (\theta) /R^3 dA ## ##F = MG\sigma \int _A b / R^3 dA## After this, I...

So from what I understand, if we drop a straight line from the center of the sphere S to where it meets the plane, say that point is O, we'll have a line SO perpendicular to the plane of length ##b##. Then if we drop another straight line from S to a particular point on the plane, say X, then...

Homework Statement This is question 3.7 from Gregory's Classical Mechanics textbook. A symmetric sphere of radius a and mass M has its center a distance b from an infinite plane containing a uniform distribution of mass ## \sigma ## per unit area. Find the gravitational force exerted on the...

Ah okay I see. In the case that ##\alpha \in \mathbb{Q}##, each ##\Phi _n## is either pairwise disjoint or exactly equal to ##\Phi _m## (for example ##\Phi _0 = \Phi _{360}## if ##\alpha = 1##, assuming we are using degree angles), so that consequently we wouldn't have the the infinite union ##C...

Hi, I'm reading through a proof of the existence of a nonmeasurable set. I've copied down the proof below more or less verbatim: In particular, I am trying to understand the significance of why ##\alpha## has to be an irrational number. Would the proof not hold if we used any other...

Homework Statement Let ##X## be a metric space with metric ##d##. Show that ##d: X \times X \mapsto \mathbb{R}## is continuous.Homework Equations The Attempt at a Solution Please try to poke holes in my proof, and if it is correct, please let me know if there's any more efficient way to do it...

Oh...I could've sworn I read somewhere earlier in the text that two topologies are defined to be different if neither is finer or coarser than the other. So I suppose "different" in this case just means they are not equal? I guess I just bashed my head over the table for an hour over...

Hi all, I'm looking for some help in understanding one of the theorems stated in section 20 of Munkres. The theorem is as follows: The uniform topology on ##\mathbb{R}^J## (where ##J## is some arbitrary index set) is finer than the product topology and coarser than the box topology; these...