I'm looking for some help in understanding the inertia tensor (not the physics, just the math).
I'm trying to figure out how to convert between the wedge product and tensor product definitions. Here's the wedge product definition $$ \mathcal I(B) = \int d^3 \mathbf x \rho(\mathbf x)\ \mathbf x \wedge (\mathbf x \cdot B)$$ where $B$ is a bivector. Thus the inertia tensor is just a linear mapping between bivectors. Because I can draw of picture of this, I understand this definition much better.
There's also the tensor product definition: $$\mathcal I = \int d^3 \mathbf x \rho(\mathbf x)\ [(\mathbf x \cdot \mathbf x)E -\mathbf x \otimes \mathbf x)]$$ where $E=\mathbf e_1 \otimes \mathbf e_1 + \mathbf e_2 \otimes \mathbf e_2 + \mathbf e_3 \otimes \mathbf e_3$. I will admit I don't understand tensors well. From what I gather $\mathbf u \otimes \mathbf v = \mathbf u \mathbf v^T$.
How do these two definitions jive with one another?