Modern Classical Physics Optics Fluids Plasmas Elasticity Relativity And Statistical Physics 1st Edition Kip S. Thorne, Roger D. Blandford - Solutions

Derive the component manipulation rules (1.9g) and (1.9h). In component notation, the inner product of two vectors and the value of a tensor when vectors are inserted into its slots are given by A.B=A; B₁, T(A, B, C) = Tijk AjBjCk› (1.9g) as one can easily show using previous equations.
The third-rank tensor S(___ , ___ , ___) and vectors A and B have as their only nonzero components S123 = S231 = S312 = +1, A1 = 3, B1 = 4, B2 = 5. What are the components of the vector C = S(A, B, ___), the vector D= S(A, ___ , B), and the tensor W = A ⊗ B?
(a) Consider a perfect fluid with density ρ, pressure P, and velocity v that vary in time and space. Explain why the fluid’s momentum density is G = ρv, and explain why its momentum flux (stress tensor) is(b) Explain why the law of mass conservation for this fluid is(c) Explain why the
(a) An electric field E exerts (in SI units) a pressure ∈oE2/2 orthogonal to itself and a tension of this same magnitude along itself. Similarly, a magnetic field B exerts a pressure B2/2μo = ∈oc2B2/2 orthogonal to itself and a tension of this same magnitude along itself. Verify that the
Use Eqs. (1.25) and (1.26) to derive the usual formulas dA = dxdy and dV = dxdydz for the 2-dimensional and 3-dimensional integration elements, respectively, in righthanded Cartesian coordinates. 2-volume = € (A, B) = ab Aa Bb=A₁B₂ - A₂ B₁ = det [A₁ A₁ B₁ A2 B2 (1.25)
Integrate the vector field A = zez over a sphere with radius a, centered at the origin of the Cartesian coordinate system (i.e., compute A · d∑).(a) Introduce spherical polar coordinates on the sphere, and construct the vectorial integration element d∑ from the two legs adθ eθ̂ and a sin
In Euclidean 2-space, let {e1, e2} be an orthonormal basis with positive volume.(a) Show that the components of ∈ in this basis are(b) Show that €12 = +1, €21= -1, €11= €22=0. (1.24a)
Convert the following equations from the geometrized units in which they are written to SI units.(a) The “Planck time” tP expressed in terms of Newton’s gravitation constant G and Planck’s reduced constant ℏ, tP =√Gℏ. What is the numerical value of tP in seconds? in meters?(b) The
Here is an example of how to use index notation to derive a vector identity for the double cross product A × (B × C): in index notation this quantity is ∈ij kAj (∈klmBlCm). By permuting the indices on the second ∈ and then invoking Eq. (1.23), we can write this asBy then invoking the
From its complete anti-symmetry, derive the four properties of the Levi-Civita tensor, in n-dimensional Euclidean space, that are claimed in the text following Eq. (1.20). volume (A, B, ..., F). = (1.20)
One of Maxwell’s equations says that ∇ × E=−∂B/∂t (in SI units), where E and B are the electric and magnetic fields. This is a geometric relationship between geometric objects; it requires no coordinates or basis for its statement. By integrating this equation over a 2-dimensional
(a) The following expressions and equations are written in slot-naming index notation. Convert them to geometric, index-free notation: AiBjk, AiBji, Sijk = Skji ,AiBi = AiBjgij .(b) The following expressions are written in geometric, index-free notation. Convert them to slot-naming index

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