By S L Sobolev

ISBN-10: 0821815571

ISBN-13: 9780821815571

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7 (a) Show that the gravitational potential of a thin uniform spherical shell centered at the origin, of radius a and total mass m, at an external point r = (0, 0, z), z > a, is unchanged when the mass of the shell is concentrated at the origin. (b) Show that the result of part (a) also holds if the spherical shell is replaced by a uniform solid sphere of radius a < z and total mass M. 5 Vector diﬀerentiation of a vector ﬁeld A vector function F(t) of a single variable t is made up of three components F x (t), Fy (t), Fz (t), each of which is a scalar function of t.

By applying Eq. 84) to each of these small closed surfaces, we ﬁnd that dσ · j(r) = S i = 1 Δτi →0 Δτi lim Ω ΔSi dσ · j(r) dτ∇ · j(r). 85) This integral relation is known as Gauss’s theorem. It states that the net outﬂow across a closed surface S is equal to the total divergence in the volume Ω inside S. As a result, we may say that the enclosed divergence “causes” the outﬂow; that is, the enclosed divergence is a “source” of the outﬂow. 86) where ε0 is the permittivity of free space. As a result, Ω dτ ρ Q = = ε0 ε0 dσ · E.

2 Calculate I = dσ · V(r) over the unit sphere for (a) V(r) = xy2 i + yz2 j + zx2 k, (b) V(r) = A(r) × r, where A(r) is irrotational. 3 Calculate the total ﬂux (or net outﬂow) over a spherical surface of radius R about the origin for the vector ﬁeld F(r) = r−a , |r − a|3 a = ai, when R < a and when R > a. 4 Verify Gauss’s theorem by showing separately that over a sphere of radius R the surface integral dσ · (r/r n+1 ) = 0 and the volume integral dτ∇ · (r/rn+1 ) are both equal to 4πR2−n . Explain why this is true even for n = 2 when ∇ · (r/rn+1 ) = 0 for all ﬁnite r.

### Applications of functional analysis in mathematical physics by S L Sobolev

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