Recent content by Odious Suspect

Thank you for taking the time to read and reply. My question has to do with the ordering of rows in the last application of Cramer's rule in the original post. ##\mathfrak{j}=\frac{\left| \begin{array}{ccc} \mathfrak{B}_1 & \mathfrak{B}_2 & \mathfrak{B}_3 \\ a_{11} & a_{12} & a_{13} \\ a_{31}...

##\hat{\mathfrak{i}}, \hat{\mathfrak{j}}, \hat{\mathfrak{k}}## are the traditional ##\hat{i}, \hat{j}, \hat{k}## of vector calculus. I use German (Fraktur) letters for vector-ish and tensor-ish things. The ##\Phi## beast is a dyadic. I am not comfortable enough with dyadics to provide a...

The following is from Donald H. Menzel's Mathematical Physics: ##\Phi =\left( \begin{array}{c} a_{11}\hat{\mathfrak{i}} \hat{\mathfrak{j}} +a_{12}\hat{\mathfrak{i}} \hat{\mathfrak{j}} +a_{13}\hat{\mathfrak{i}} \hat{\mathfrak{k}} \\ +a_{21}\hat{\mathfrak{j}} \hat{\mathfrak{i}}...

This is an observation, not a criticism. I noticed that in The Feynman Lectures on Physics Vol-I, Chapter 9 Newton's Laws of Dynamics in section 9.5 he speaks of "the law of dynamics" in the singular "That is where the law of dynamics comes in. The law of dynamics tells us what the...

But I'm not "allowed" to think in terms of Taylor series. It hasn't been introduced yet.

I'm not sure I'm parsing this correctly. Is this what you intended? ##y_{n+1}-y_n(x)y(x)##. And do you mean for me to leave ##y(x)## as an symbolic, yet to be determined function? Darn! Library is closing. Will pick this up later.

I guess I don't follow. I have a generic form for the nth term ##\int_0^x 1 \, df_n(t)=2 \sum _{k=2}^{n-1} \frac{x^k}{k!}+\frac{x^n}{n!}+x+1##, but I'm not sure why I should assume it is a better approximation than its predecessor. Sure, I can carry out the process to produce an infinite...

This is from an example in Thomas's Classical Edition. The task is to find a solution to ##\frac{dy}{dx}=x+y## with the initial condition ##x=0; y=1##. He uses what he calls successive approximations. $$y_1 = 1$$ $$\frac{dy_2}{dx}=y_1+x$$ $$\frac{dy_3}{dx}=y_2+x$$ ...

Sorry about being so abrupt. I get grouchy when I can't figure something out. I'm guessing this might be something to do with rotation of coordinates to eliminate the "cross product" term. In some universe, the denominator in the cosine expression represents a radius.

That only addresses the form I really am familiar with. Notice that the form I am asking about is more complicated.

No. I can prove it. I want a geometric development. That is to say, I believe the constants and variables in the expression ##cos \theta = \frac{b-hx}{\sqrt{a+b^2h}}## have some geometric significance pertaining to an ellipse.

I posted a question about this yesterday, but realized I had made a stupid mistake in my derivation. Orbital dynamics: "The familiar arc-cosine form" That error has been corrected. I still have a deeper question. I believe this expression can be developed using the geometry of an ellipse in...

Whoops! ##\frac{du}{dx}=\frac{-h}{\sqrt{a+b^2h}}##. I guess I have to embarrass myself before I can find my mistakes. I think that correction will get me most of the way there. I would still like to understand where that "familiar form" came from. That is, how is it arrived at "working...

This arises in Joos's discussion of planetary motion, at the following URL: https://books.google.com/books?id=wFl2MkpcY6kC&lpg=PP1&pg=PA91#v=onepage&q&f=false I've modified the notation in obvious ways. He asserts that the following expression is "the familiar arc-cosine form": $$-\int...

As I understand things, projecting onto orthonormal basis vectors is the resolution of the vector into components. That's why I prefer to speak of the signed volume being linear in each of the components of its arguments, rather than in the arguments themselves. Now, we might argue that the...