Recent content by BloonAinte

Thank you for your comment. I understand what you have said. Apologies to necrobump a thread with a wrong solution. This is actually the official solution I was given. (For future readers' notice: The correct version of this result is proved in Post #7 and and also be found in Wikipedia, under...

For closure on this, I wanted to share the solution to the intended problem in Post 14. Consider the field ##a \times F##. Applying Stoke's theorem gives ##\int_S \nabla \times (a \times \mathbf{F}) \cdot d\mathbf{S} = \int_C (a \times \mathbf{F}) \cdot d\mathbf{r}## Then, ##(a \times...

Thank you so much for all your help! :) I understand this more now ^^

Thank you! Does this method always determine the first shock? I have looked up an example to illustrate my doubt: https://math.stackexchange.com/questions/2951455/determine-the-shock-curve-and-sketch-characteristics-in-xt-plane That question considers some PDE and obtains characteristics as...

To find a shock wave, do we always solve the equation ##x_{\xi}=0##? The PDEs I consider are of the form ##u_t + g(u) u_x = f(u)##, with initial condition ##u(x,0) = h(x)##. I have been looking at the solutions for problems in my homework sheet but this method was used with no explanation. Why...

Thank you so much!

I woud like to find the characteristic curves for ##u_t + (1-2u)u_x = -1/4, u(x,0) = f(x)## where ##f(x) = \begin{cases} \frac{1}{4} & x > 0 \\ \frac{3}{4} & x < 0 \end{cases}##. By using the method of chacteristics, I obtain the Charpit-Lagrange system of ODEs: ##dt/ds = 1##, ##dx/ds = 1 -...

Thank you! I chose a surface with ##dS = (0,0,1)##, ##b=(1,2,0), c=(1,2,0)## so that ##F=(1(x+2y), 2(x+2y),0)##. Then ##\nabla \times (F \times dS) = (0,0,-5).## However, ##(dS \times \nabla) \times F = (0,0,0)##. Here ##dS \times \nabla = (- \partial_y, \partial_x, 0)## and thus ##(dS \times...

Thank you for your hint. Apologies to be a bother but I wasn't able to obtain the required field. For ##F = (x^2 ,0,0)## it does not hold with a = (1,1,1) but the integrals agreed For ##F = (e^x, e^y, 0)##, it does not hold with a = (1,1,1) but I obtain 2e - 2 for both integrals when...

Thank you, but I tried again using ##F(x,y,z) = (x^2,0,0)## in the second part of the post, which still gave the answer. Do you have a particular suggestion?

Hi! I just wanted to write a quick correction for future readers. Note that what I did in post 14 is wrong. The mistake arose from wrongly using the cross product in cylindrical coordinates. To remedy this, instead use Cartesian coordinates. We write ##F(x,y,z) = (x,y,0)## and consider a circle...

Thank you. I did a misread and in fact it is the expression in #14 which was in the question.

Hi, I've been exploring this a bit more. Would the identity work if I write ##\int_S \nabla \times (F \times dS)##? I considered the circle of radius r in the xy plane, and I got ##\nabla \times (F \times dS) = (0,0,-4r \cos \theta)dr d\theta## which integrates to ##(0,0,0)## unlike the ##(0,0,2...

Wow, that's really cool! Do you have a reference for this? I'd like to learn more about this. Does this relate to the study of differential forms/generalized Stokes?

I understand. Thank you for all your help! I very much appreciated your hints and how you helped me reach the answer.