Recent content by leonardthecow

Ah okay, I buy that, thanks! Related question though; ##\vec{E_0}## is defined as $$\vec{E_0}=\textbf{A}_1 + i\textbf{A}_2,$$ where ##\textbf{A}_2## and ##\textbf{A}_2## are in ##\mathbb{R}^3##. In a later proof, my professor makes the claim that $$\vec{k} \cdot \textbf{A}_1 = \vec{k} \cdot...

Homework Statement This isn't necessarily a problem, but a question I have about a certain step taken in showing that the electric and magnetic fields are transverse. In Jackson, Griffiths, and my professor's written notes, each claims the following. Considering plane wave solutions of the...

Hey All, I recently posted this in another area but was suggested to put it here instead. Here is my original post:

Homework Statement Hi all, I'm currently reviewing for a final and would like some help understanding a certain part of this particular problem: Determine the retarded Green's Function for the D'Alembertian operator ##D = \partial_s^2 - \Delta##, where ##\Delta \equiv \nabla \cdot \nabla## ...

Ah thank you! That's the step I was missing, 3 is a straight line. This is exactly what I needed.

Well, the original problem is to compute the line integral of $$v= r\cos^2(\theta)\hat{r} -r\cos(\theta)\sin(\theta)\hat{\theta} +3r \hat{\phi}$$ around a path depicted in the text. The path, in terms of spherical coordinates, runs as follows, broken into 3 segments: 1) $$r:0 \rightarrow 1...

Homework Statement Hi there! So I'm working on an old homework problem for review so that I actually have the solution, but I'm not sure how to compute a certain part. Here it is: $$\int \int rcos^2(\theta)dr - r^2cos(\theta)\sin(\theta)d\theta$$ Homework Equations The solution involves (what...

So I tried parts for the first one, and assuming I didn't mess up somewhere, if I choose u to be the exponential term then I end up with $$e^{-x^2}\frac{x^5}{5} - \int^{\infty}_{-\infty}\frac{x^5}{5}(-2xe^{-x^2})$$, with the first term being evaluated from minus infinity to infinity as well...

Hey everyone, I'm wondering how to solve the following definite integral, \int^\infty_{-\infty}{x^4e^{-x^2}dx}. I know the answer is ##\frac{3 \sqrt{\pi}}{4}##, but I'm not positive how to get there. I understand how to evaluate the definite "Gaussian" integral...

Wow that a struggle...I really appreciate all the guidance and persistence though, it definitely helped more than the textbook or lecture did. Thank you!

Here goes... Let ∑fn(x) from n=1 to n=∞ be a series of functions all defined for a set of values of x. If there is a convergent series of constants ∑Mn from n=1 to n=∞, such that |fn(x)| ≤ Mn for all x in the set of values of x, then the series is uniformly convergent in the set of values of...

The convergence of that series of functions would have to be less than or equal to 1/n2 for x in the interval [2,∞), which is true as shown by the result of the p-test since the p-test shows that the series converges for x>1, which intersects the set [2,∞)...right?

Ah I'm sorry, I misread that. Well the maximum for a fixed n would then be 1/n2, since the value of the function would decrease as the denominator gets larger with x

Well, the function decreases as n gets larger, so the largest value it can take will be 1/2x, assuming x is positive

Mn is just a number, correct? And (maybe this is my point of non understanding then) we're assuming for the sake of the test that |fn(x)| ≤ Mn, so the previous statement is true by definition? I realize that logic is more or less circular, where I'm assuming what I'm trying to prove in order...