Aris' book Vectors, Tensors, and the Basic Equations of Fluid Mechanics describes how to convert between covariant, contravariant, and physical components of vectors and tensors.
For example, in chapter 7 he defines the covariant metric tensor as: $$g_{ij} = \sum_k \frac{\partial y^k}{\partial x^i}\frac{\partial y^k}{\partial x^j},$$ where $y^i$ are Cartesian coordinates and $x^i$ are curvilinear coordinates. The contravariant metric tensor is defined through the identity: $$g^{ij}g_{jk} = g_{kj}g^{ji} = \delta_i{}^k = \delta^i{}_k.$$
He then uses these metric tensors to convert covariant and contravariant vectors and tensors to their physical components. For example the physical components of $\mathbf{A}$ are denoted by $A(j)$: $$A(j) = \sqrt{g_{jj}}A^j\qquad \text{(no summation on $j$)}$$
For orthogonal coordinates, where $h_i = \sqrt{g_{ii}}$, the conversion of tensors to physical components is: $$T(ij) = \frac{h_i}{h_j}T^i{}_j\qquad \text{(no summation on $i$ or $j$)}$$ $$T(ij) = \frac{h_j}{h_i}T_i{}^j\qquad \text{(no summation on $i$ or $j$)}$$ $$T(ij) = h_ih_jT^{ij}\qquad \text{(no summation on $i$ or $j$)}$$ $$T(ij) = \frac{1}{h_ih_j}T_{ij}\qquad \text{(no summation on $i$ or $j$)}$$ He then says that for non-orthogonal coordinates the form is similar to the first one above: $$T(ij) = \sqrt{\frac{g_{ii}}{g_{jj}}}T^i{}_j\qquad \text{(no summation on $i$ or $j$)}$$ and you can determine the conversion of other tensors by raising/lowering the indices of the tensor on the right-hand-side using the metric tensor, such as: $$T(ij) = \sum_m\sqrt{\frac{g_{ii}}{g_{jj}}}g^{im}T_{mj}\qquad \text{(no summation on $i$ or $j$)}$$
This explanation is sufficient for me to determine these conversions for all second-order tensors. How do I determine the physical components of a third-order tensor?
$$T(ijk) = ?$$
EDIT:
Aris provides this description of physical components:
As you have already pointed out one can compute the physical component of a contravariant vector using $$A_{(i)} = h_{(i)} A^i$$ For a covariant vector we have $$A_{(i)} = h_{(i)} g^{i j} A_j \overset{\text{orthogonal coordinates}}{=} \frac{1}{h_{(i)}} A_i$$ This "transformation" can also be extended to tensor of arbitrary rank $$ T_{(i_1,..,i_n,j_1,..,j_m)} = \frac{h_{(i_1)}\cdot ... \cdot h_{(i_n)}}{h_{(j_1)}\cdot ... \cdot h_{(j_n)}} {T^{i_1,..,i_n}}_{j_1,..,j_m}$$