Recent content by Ben2

Using either H&R's Chapter 27 Example 3 or Problem 590 of the ##\mathbf{Physics Problem Solver}##, I've been unable to get the component ##E_x## or ##E_y##. There are now different angles at the charges. My thanks to berkeman for LaTeX advice, but any errors are of course my own. Thanks in...

Thanks to all for the references.

Thanks very much.

For F: X x I-->Y, defined by F(x,t) = y, next define G: Y x I-->X by G(y,u) = x. Then for t = u, we have F[G(y,t),t] = F{G[F(x,t),t]}, which will ideally be ##\mathbb{1}##. Given Hatcher's definitions pp. 2-3, to me it's not clear how to "invert" a homotopy without an inverse function--let...

Thanks for the comments. For Mentor berkeman: Pending a computer cleanup, I've held off on installing a LaTeX translator (Is MathJax what you want?). On uploading diagrams, I'm a complete dunce. For haruspex: That is all correct. For kuruman: Besides the form of H&R's second equation p...

Showing the motion is simple harmonic seems routine. The 5th equation on p. 674 gives ##E=frac{1}{4\pi\epsilon_0}frac{qx}{(a^2)+(x^2)}^frac{3}{2}##, but matching expressions for ##\omega=k/m## yields only ##x=frac{ea^2}{2}##. Something in the model is escaping me. Thanks for any help offered!

Apologies for posting! With the conditions given, (hf)(gk) = h(fg)k\congh(\mathbb{1})k = hk\cong\mathbb{1} and (fh)(kg) = f(hk)g\congf\mathbb{1}g = fg\cong\mathbb{1}. This handles transitivity, while the reflexive and symmetric properties are routine. Thanks to everyone who read this!

"[A] map f: X-->Y is called a \mathbf{homotopy~equivalence} if there is a map g: Y-->X such that fg\cong\mathbb{1} and gf\cong\mathbb{1}," where "cong" means "is homotopic." "The spaces X and Y are said to be \mathbf{homotopy~equivalent}..." Additional definitions are in Hatcher, "Algebraic...

Will study the whole thread, and thanks to all respondents! "Anticipating" later developments is common in math, but I'd not seen enough to realize it happens in physics.

I've never been able to scan a document picture. Here haruspex draws the same conclusion as I do. TSny's comment is also helpful, in the sense H&R's problem editor seems to have made additional assumptions, at least on q. I smelled E as a function of x and y. But I've never used vectors in...

For lines of force symmetric with respect to the angle bisector of "near" tangent lines: An adaptation of the figure suggests that a right triangle with hypotenuse parallel to the right-hand "far" tangent line is similar to the right triangle with hypotenuse parallel to the "near" tangent line...

Thanks to Tsny and pasmith for help with this! Will do Problem 18.20 as suggested. Ben2

Thanks for your timely response! I've not previously heard of a velocity gradient. Figure 18-20 features ten horizontal streamlines, where the spacing narrows from top and bottom to the middle three. Theorem 10, Chapter 13 of Stewart's "Calculus" gives the curvature k(t) = |r'(t) x...

I tried using these equations, but it's not clear if we should hold y_1 = y_2. A transverse velocity vector would produce a flow at some angle to the horizontal, but How do they known there's such a vector?

Good suggestion! I've often wondered if different radial sections of a hinge could experience different amounts of friction.