A Guided Tour of Mathematical Physics by Roel Snieder

By Roel Snieder

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2) to derive the theorem of Stokes. Problem a: Compute the flux of the vector field v(x, y, z) = (x + y + z)ˆ z through a sphere with radius R centered on the origin by explicitly computing the integral that defines the flux. Problem b: Show that the total flux of the magnetic field of the earth through your skin is zero. Problem c: Solve problem a without carrying out any integration explicitly. 1) to show that the gravitational field of a body with a spherically symmetric mass density ρ depends only on the total mass but not on the distribution of the mass over that body.

626 × 10−34 kg m2 /s. Suppose we are willing to accept that the motion of an electron is described by the Schr¨odinger equation, then the following question arises: What is the position of the electron as a function of time? According to the Copenhagen interpretation of quantum mechanics this is a meaningless question because the electron behaves like a wave and does not have a definite location. Instead, the wavefunction ψ(r, t) dictates how likely it is that the particle is at location r at time t.

The corresponding volume integral V p1 f2 dV is in that case given by p1 (r0 ). The wavefield p2 (r) generated by this point source is called the Green’s function, this special solution is denoted by G(r, r0 ). (The concept Green’s function is introduced in great detail in chapter 14. ) The argument r0 is added to indicate that this is the wavefield at location r due to a unit source at location r0 . e. f1 = 0). Let us simplify the notation further by dropping the subscript ”1” in p1 . 10) can be written as: p(r0 ) = S 1 (p(r)∇G(r, r0 ) − G(r, r0 )∇p(r)) · dS .

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