# Solutions Of Exercises Of Principles Of Tensor ...

The classical proofs of the parabolic Harnack inequality do not give particularly sharp bounds on the constant . Such sharp bounds were obtained by Li and Yau, especially in the case of the scalar heat equation (1) in the case of static manifolds of non-negative Ricci curvature, using Bochner-type identities and the scalar maximum principle. In fact, a stronger differential version of (3) was obtained which implied (3) by an integration along spacetime curves (closely analogous to the -geodesics considered in earlier lectures). These bounds were particularly strong in the case of ancient solutions (in which one can send ). Subsequently, Hamilton applied his tensor-valued maximum principle together with some remarkably delicate tensor algebra manipulations to obtain Harnack inequalities of Li-Yau type for solutions to the Ricci flow (2) with bounded non-negative Riemannian curvature. In particular, this inequality applies to the -solutions introduced in the previous lecture.

## Solutions of Exercises of Principles of Tensor ...

If you have additional exercises that would be appropriate for ModernProblems in Classical Electrodynamics, and would be willing to sharethem, e-mail them to me at the address below. I will be more than pleasedto post them on this web site and give you credit for submitting them. Solutionswould be useful, too! (I won't post the solutions.)

The day's discussion focussed on symmetries: mainly on Noether's theoremand the topic of tensors and group representations. As follow up exercisesRead about the field due to static charges and their multipole expansions from Landau and Lifschitz sections 40 and 41 and Jackson sections 4.1 and 4.2. The former book exhibits the tensor character in terms of multi-index objects, the latter in terms of irreducible representations (irreps) of the rotation group through the spherical harmonics.Learn about rotational tensors and irreps of the rotation group from the appendix on this subject in "Nuclear Physics" by De Shalit and Talmi.

Follow up questionsWrite the stress tensor as a sum of symmetric and antisymmetric parts. What is the contribution of the antisymmetric part to the angular momentum?For plane waves propagating in the +x direction, it was demonstrated that it is possible to choose a Lorentz gauge in which the components A0 and A1 of the gauge potential vanishes. Is it possible to make more components vanish?What condition must Lorentz transformations satisfy if they are to diagonalize the stress tensor for plane wave solutions of the Maxwell equations? What Lorentz transformations, if any, satisfy these conditions?Construct possible actions, the equations of motion, consider the role of symmetries, the stress tensor and plane wave solutions for a real scalar field φ a complex scalar field ψ=ψR+iψI, such that the action is invariant under changes of phase, ie, for ψ'=exp(iα)ψ, where α is a constant, independent of the position. M different scalar fields, φ1 ... φM such that the action is invariant under linear combinations of field components which are the same at each space-time point. 041b061a72