Recent content by matphysik

I was not able to correct my rough work above, so i shall do it now. You (`ozone`) say that, "I can't figure out how it was DERIVED". The `derivation` follows from simple multivariable calculus. Given dy/dx=-x/2y, rewrite it as 2ydy+xdx=0. Next, introduce F∈C¹(ℝ²) and set F(x,y)=constant...

If `a` and `b` are parallel then ∃c∈ℝ s.t., a=cb. So then, ||a+b||=||cb+b||=||b||(|c+1|)≠||a||+||b||. Since ||a||=||cb||=(|c|)||b||.

Set F(x,y)≡-x/2y then 0=dF=(∂F/∂x)dx+(∂F/∂y)dy and, -(∂F/∂x)/(∂F/∂y)=dy/dx. Reversing the steps should answer your question.

1. Given, s(t)=4t². You seek the `instantaneous speed` when t=3 by way of first principles: Let s(t+δt) and δt be increments of the variables `s` and `t`, respectively. Consider, s(t+δt)-s(t)=4(t+δt)²-4t²=4[2tδt+δt²]. Let ds/dt≡ lim [s(t+δt)-s(t)]/δt as δt→0 denote the `instantaneous speed`...

Which type of integral is implied in your o.p.?

Hello. s=4t² ⇒ ds/dt=8t. Hence, ds/dt|(t=3) = 8(3)= 24.

In 1D: F·dx=ma·dx=mdv/dt·dx=mdv·dx/dt=mvdv⇒∫F·dx=½mv²+constant. Similarly for the `impulse`, F·dt=ma·dt=mdv/dt·dt=mdv⇒∫F·dt=mv+constant.

Q and P are defined for all ψ∈S⊃φ (not φ).

Is he a mathematician?

Let S denote the space of "rapidly decreasing functions", then (on top of that already mentioned in the o.p.) we have the following string of set inclusions: φ=D(G) ⊂ S ⊂ H=L₂(G) ⊂ φˣ=D`(G).

Correction: b² - a²=9 ⇒ b=√13. So that, ℐ= ∫ ½du/√(9-u) - 2sin⁻¹[(x+2)/√13] + constant.

Gravitational Time Dilation: The proper time measured by a clock moving with 3-velocity vᵃ= dxᵃ/dt (a=1,2,3.) in a spacetime with metric gᵤᵥ (u,v=0,1,2,3.) is given by: dτ = √(- c⁻ ² gᵤᵥdxᵘdxᵛ)·dt= √(-g₀₀ - 2gₐ₀ dxᵃ vᵃ/c - v²/c²)·dt. Where v²=gₐₑvᵃvᵉ (a,e=1,2,3.). For v=0, dτ =√(-g₀₀)·dt...

Repulsive and attractive `forces` is Newtonian theory(!).

Yes, that`s aside of what i said. What you`re saying follows immediately from the interpretation of |ψ(·,t)|² as the probability distribution of position in x-space, with its momentum analogue in k-space. Where ψ∈L₂, is the so-called `wave function`.

φ=D(G) H=L₂(G) and φˣ=D`(G), where G is a nonempty open set of ℝⁿ with n≥1. D is short for the space of continuously differentiable functions with compact support, and D` is the space dual to D. REMARK: For u∈D, u:G→ℂ