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?

• Welcome to SE.Physics! You may want to give a specific example in your question statement to help focus the problem. The specific issue is that it's not entirely clear what you might mean by a tensor's magnitude in a particular direction; for example, are you asking how to identify the components within a tensor? Whatever the case, if you provide a specific example, it can help clarify what you mean to ask. – Nat Aug 6 at 13:40

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 $$T_{r\theta}$$.

• Yes, that is what I wanted. How to calculate the component in a particular direction. – user49535 Aug 6 at 13:14
• @user49535: If you want to know how to calculate a component of the tensor, you need to specify what information you already have about the tensor. – Ben Crowell Aug 6 at 15:57

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.

• Thanks a lot !! Will need some time to soak it in. Can you please suggest any reference book regarding this topic. Still have few things not clear. Specially, how you got the two components (cosθcosϕ sinθsinϕ −sinθ) and (−sinθ Cosθ 0). – user49535 Aug 7 at 21:43
• Those are the polar basis vectors $\hat{\phi}$ and $\hat{\theta}$ expressed in terms of the Cartesian basis vectors, i.e. $\hat{\phi} = \cos \theta \cos \phi i + \sin \theta \sin \phi j - \sin \theta k$ which I have represented with the column matrix. A quick google revealed that to me. An elementary introduction to tensors is given here in Chapter 26 and a more advanced mathematical explanation is given here in chapter 2. – Matt0410 Aug 8 at 22:48