What is the magnitude of a tensor property in a fixed direction? If I have a physical property represented by a $3 \times 3$ tensor, how can I find its magnitude in a particular direction, say $(\phi, \theta)$ in spherical coordinate system?
 A: A tensor doesn't have a magnitude, and it doesn't have a component along a particular direction. Those are properties of vectors. A tensor in three dimensions has 9 components, each of which corresponds to two directions. For example, in spherical coordinates, a tensor $T$ could have a component $T_{r\theta}$.
A: In this context, a tensor is an abstract operator on Euclidean space $T : \mathbb{R}^3 \rightarrow \mathbb{R}^3$. In other words, a tensor takes in a vector $\mathbf{v} \in \mathbb{R}^3$ and maps it to another vector $\mathbf{v}' = T(\mathbf{v}) \in \mathbb{R}^3$. The components of a vector depend on which basis we are using, and the same applies to a tensor.
In general, if we have a basis $(e_1,e_2,e_3)$ of Euclidean space, the components of the tensor are defined as
$$ T_{ij} = e_i \cdot T(e_j) $$
where $ \cdot$ is the dot product. For example, suppose we had a tensor $T$ and in the Cartesian basis $(i,j,k)$ it has the components 
$$ M(T)= \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$
where $M(T)$ is the matrix of components of $T$, and is a representation of $T$.
The $T_{xy}$ component would be 
$$ T_{xy} = i \cdot T(j) = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix}\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = b $$ 
as expected. Now suppose we work in spherical polar coordinates with basis $( \hat{r} , \hat{\theta},\hat{\phi})$, we would have 
$$ T_{\phi \theta} = \hat{\phi} \cdot T(\hat{\theta}) \\
= \begin{pmatrix} \cos \theta \cos \phi & \sin \theta \sin \phi & - \sin \theta \end{pmatrix} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \begin{pmatrix} -\sin \theta \\ \cos \theta \\ 0 \end{pmatrix} = \ldots$$
In general, if you know the components $T_{ij}$ in one coordinate system $(x^1,x^2,x^3)$ and you want to find out what its components $T_{ab}$ are in another coordinate system $(y^1,y^2,y^3)$, you use the formula
$$ T_{ij} = \sum_a \sum_b \frac{\partial y^a}{\partial x^i} \frac{\partial y^b}{\partial x^j} T_{ab}  $$
where $y = y(x)$ is the equation for the change of coordinates.
