Recent content by smoking-frog

Those are just the coefficients, though, aren't they? ##\big( x^i \partial_i = x'^j \partial'_j \big)## does not imply ##x^i=x'^i##.

Are you sure? I see no assumption that the transformation need be linear in the derivation of this transformation.

$$x'=x'^\nu \frac{\partial}{\partial x'^\nu}$$ Then $$ \frac{\partial}{\partial x'^\nu}=\frac{\partial x^\mu}{\partial x'^\nu}\frac{\partial}{\partial x^\mu}\\ x'^\alpha=\frac{\partial x'^\alpha}{\partial x^\beta}x^\beta$$ Then $$x'=x'^\nu \frac{\partial}{\partial x'^\nu}=\frac{\partial...

Why is ##x^\mu \partial_\mu## not a vector field?

Thanks! Yeah, I forgot the basis vectors in the metric, that was a typo.

Proposition: Consider an ##n + 1##-dimensional metric with the following product structure: $$ g=\underbrace{g_{rr}(t,r)\mathrm{d}r^2+2g_{rt}(t,r)\mathrm{d}t\mathrm{d}r+g_{tt}(t,r)\mathrm{d}t^2}_{:=^2g}+\underbrace{h_{AB}(t,r,x^A)\mathrm{d}x^A\mathrm{d}x^B}_{:=h} $$ where ##h## is a Riemannian...

Hey! How to transform the equation \bigtriangleup\vec E=\operatorname{div}(\operatorname{grad}(\vec E))=\epsilon_0\cdot\mu_0\cdot\frac{\partial^2\vec E}{\partial t^2} in Einstein Notation? Thank you all for your help!

Homework Statement Write \nabla \times (\nabla \times \vec A) in Einstein-Notation, whereas \vec A is the vector potential of the magnetic field. Homework Equations (\vec a \times \vec b)=\varepsilon_{ijk} a_j b_k The Attempt at a Solution \nabla \times (\nabla \times \vec...

Oh ok. I actually tried that already, but the result was 0 for some reason... But it's good to know I had the right idea and obviously just got lost in the math on the way. I guess I'll simply try it again :) Thanks for your help!

But they are not scalar, are they? \vec d \text{ and } \vec R are vectors... The rule for vectors is, in my opinion, \operatorname{grad}(\vec a \cdot \vec b)=(\vec a \cdot \nabla) \cdot \vec b+(\vec b \cdot \nabla) \cdot \vec a+\vec a \times (\nabla \times\vec b)+\vec b \times (\nabla...

Homework Statement Calculate the Electric field of a Dipole from its Potential. \vec E=-\operatorname{grad}(\Phi_D) Homework Equations \Phi_D(\vec R)=\frac{Q}{4\pi\epsilon_0} \cdot \frac{\vec d \vec R}{R^3} The Attempt at a Solution Hi all! I am trying to calculate the electric Field of a...