Chapter 10: Tensor Analysis

Use the results of Problem 1to find the velocity and acceleration components in spherical coordinates. Find the velocity in two ways: starting with ds and starting with$\mathit{s}\mathbf{=}\mathit{r}{\mathit{e}}_{\mathbf{r}}$.

Carry through the details of getting $\mathbf{(}\mathbf{7}\mathbf{.4}\mathbf{)}$from$\mathbf{(}\mathbf{7}\mathbf{.2}\mathbf{)}$ and $\mathbf{(}\mathbf{7}\mathbf{.3}\mathbf{)}$ . Hint: You need the dot product of $\mathit{e}{\mathbf{\text{'}}}_{\mathbf{\beta }}$and $e_j$ . This is the cosine of an angle between two axes since each eis a unit vector. Identify the result from matrixAin$\mathbf{(}\mathbf{2}\mathbf{.10}\mathbf{)}$ .

Consider the matrix A in$\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{7}\mathbf{)}\mathbf{}\mathit{o}\mathit{r}\mathbf{}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{10}\mathbf{)}$ .Think of the elements in each row (or column) as the components of a vector. Show that the row vectors form an orthonormal triad (that is each is of unit length and they are all mutually orthogonal), and the column vectors form an orthonormal triad.

As we did in (3.3) , show that the contracted tensor ${\mathit{T}}_{\mathbf{i}\mathbf{i}\mathbf{j}}$is a first-rank tensor, that is, a vector.

Using cylindrical coordinates write the Lagrange equations for the motion of a particle acted on by a force, where V is the potential energy. Divide each Lagrange equation by the corresponding scale factor so that the components of F (that is, of) appear in the equations. Thus write the equations as the component equations of, and so find the components of the acceleration a. Compare the results with Problem.

For Example 1, write out the components of U,V, and $\mathit{U}\mathbf{\times }\mathit{V}$in the original right-handed coordinate system and in the left-handed coordinate system S' with the axis reflected. Show that each component of$\mathit{U}\mathbf{\times }\mathit{V}$inS'has the “wrong” sign to obey the vector transformation laws.

Divide equation (10.4) by dt to show that the velocity $\mathbf{v}\mathbf{=}\frac{\mathrm{ds}}{\mathrm{dt}}$is a contravariant vector. Note that the contravariant components of the velocity in polar coordinates are and (not r˙ and r ˙ θ which are physical components). As we did in (10.11), write the velocity v in polar coordinates in terms of the unit e vectors and in terms of the covariant a vector. Repeat the problem in spherical coordinates

As in Problem 2, complete Example 5.

IfE= electric field andB= magnetic field, is $\mathit{E}\mathbf{\times }\mathit{B}$ a vector or a pseudovector? Comment: is called the Poynting vector; it points in the direction of transfer of energy. Does that tell you from the physics whether it is a vector or a pseudovector?

Show that the contracted tensor ${\mathit{T}}_{\mathbf{i}\mathbf{j}\mathbf{k}}{\mathit{V}}_{\mathbf{k}}$is a ${\mathbf{2}}^{\mathbf{n}\mathbf{d}}$-rank tensors.