Recent content by ansrivas

Look at the definition of a line integral in this case and also note that curl (grad \phi) =0 so grad \phi defines a vector field which is conservative. Look at how this links to path independence and defines a consistent \phi

As anticipated some algebraic mistakes :) \int{z}^{2}{dz} = \int^{\pi}_{0}(sin\theta - 4sin\theta\cos^{2}\theta)d\theta + i\int^{\pi}_{0}(cos\theta - 4cos\theta\sin^{2}\theta)d\theta \int{z}^{2}{dz} = \left[- cos\theta + \frac{4}{3}cos^{3}\theta\right]^{\pi}_{0} + i\left[sin\theta -...

I have been reading this book on and off for at least 6 months now. I was trying to get through the chapters on Calculus on Manifolds (the last I remember:)). I think this book is very different from other books on modern physics that have been written for people not working in physics/maths...

I tried to be clear that we are not talking about test charges here at all. The issue is we all know that charge is quantized. So what does one exactly mean by volume density of charge. This is made clear by the question here where we are discussing the field at the very point we have an...

I think what the issue is trying to get at is that if we calculate the electric field of a charge distribution in matter then what happens to E at the exact position where we have the charge (just based on classical electrodynamics).

It is a current and will result in a magnetic field. Current is the flow of charge in space. In the frame of the ring there is no current.

Electric and Magnetic fields in matter are the macroscopic fields ... which means the average over regions large enough to contain many atoms ... The actual microscopic fields will fluctuate strongly inside matter ... This is what I understand from Griffiths.

a) Let (x0, y0) be a point on the curve. Then there exists t0 such that x_0=\frac{3t_0}{1+t_0^{3}} y_0=\frac{3t_0^{2}}{1+t_0^{3}} Now what's the point for t=1/t0 .. that is x_1=\frac{3/t_0}{1+1/t_0^{3}}=y_0 y_1=\frac{3/t_0^2}{1+1/t_0^{3}}=x_0 E) Solve for t by...

For Question 1 I believe its false. Eg. f(x)=1 if x is rational 0 o.w.

I=\int \frac{\sin \theta - \cos \theta}{(\sin \theta + \cos \theta)\sqrt{\sin \theta \cos \theta + \sin^2 \theta \cos^2 \theta}} d \theta = \int \frac{\sin^2 \theta - \cos^2 \theta}{(1 + 2\sin \theta \cos \theta)\sqrt{\sin \theta \cos \theta + \sin^2 \theta \cos^2 \theta}} d \theta Now let...

Thanks yip. I was stumped on that one for some time.

Another interesting integral I remember from one of Feynman popular book is \int_0^\pi \log\left( 1-2\alpha \cos x + \alpha^2 \right) dx However, I can't recall it at all now. I remember it was the same trick as that used to find the definite integral of the sinc function (differentiate a...

I=\int_0^\infty \exp\left( -i x^k \right) Substituting: u=ix^k x\frac{du}{dx}=kix^k=ku I=\left( \int_0^\infty \exp(-u)u^{1/k-1} \right) \frac{(-i)^{1/k}}{k} =\frac{1}{k}\Gamma[\frac{1}{k}]\exp\left( \frac{-i\pi}{2k}\right) = \Gamma[(k+1)/k]\exp\left( \frac{-i\pi}{2k}\right) Comparing...

I=\int^1_0 \frac{\log(1+x)}{x}\,dx = \int^1_0 \frac{1}{x}\sum_{i=1}^{\infty}\frac{x^i(-1)^{i+1}}{i} =\int_1^0 \left( 1 - \frac{x}{2} + \frac{x^2}{3} - \ldots \right)\,dx = 1 - \frac{1}{2^2} + \frac{1}{3^2} - \ldots Using \sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6} also...

1) ||Ax|| is equal to the component of y=Ax with the maximum magnitude. Now each component of y satisfies y_i = A_i^Tx \leq \sum_j |a_{ij}| for ||x||_{\infty} \leq 1 So the infinity norm of a matrix is \max_{i} \sum_j |a_{ij}| Now it is straightforward to identify the desired...