To solve $\mathbf{u} \cdot \mathbf{T} - \mathbf{T}^\top \cdot \mathbf{u}$ using basis vectors and basis dyadics, we will expand it step-by-step.

Step 1: Represent the Vectors and Tensors in Basis Form

Let:

• $\mathbf{u}$ be a vector represented as $\mathbf{u} = u_i \mathbf{e}_i$ in terms of its components $u_i$ and the basis vectors $\mathbf{e}_i$.
• $\mathbf{T}$ be a second-order tensor represented as $\mathbf{T} = T_{ij} \mathbf{e}_i \otimes \mathbf{e}_j$, where $T_{ij}$ are its components in the basis.

The transpose of $\mathbf{T}$ is given by: $\mathbf{T}^\top = T_{ji} \mathbf{e}_i \otimes \mathbf{e}_j$

Step 2: Expand Each Term

1. Term 1: $\mathbf{u} \cdot \mathbf{T}$: $(\mathbf{u} \cdot \mathbf{T}) = u_i \mathbf{e}_i \cdot (T_{jk} \mathbf{e}_j \otimes \mathbf{e}_k)$ Using the dot product $\mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij}$, we simplify: $(\mathbf{u} \cdot \mathbf{T}) = u_i T_{ij} \mathbf{e}_j$

2. Term 2: $\mathbf{T}^\top \cdot \mathbf{u}$: $(\mathbf{T}^\top \cdot \mathbf{u}) = (T_{ji} \mathbf{e}_i \otimes \mathbf{e}_j) \cdot u_k \mathbf{e}_k$ Again, using the dot product, this becomes: $(\mathbf{T}^\top \cdot \mathbf{u}) = u_k T_{ji} \delta_{jk} \mathbf{e}_i = u_j T_{ji} \mathbf{e}_i$

Step 3: Subtract the Two Terms

The subtraction of the two terms is: $(\mathbf{u} \cdot \mathbf{T}) - (\mathbf{T}^\top \cdot \mathbf{u}) = (u_i T_{ij} \mathbf{e}_j) - (u_j T_{ji} \mathbf{e}_i)$

Breaking this into components: $(\mathbf{u} \cdot \mathbf{T} - \mathbf{T}^\top \cdot \mathbf{u}) = u_i T_{ij} \mathbf{e}_j - u_j T_{ji} \mathbf{e}_i$

Step 4: Component Interpretation

The result is a vector, where the $j$-th component is given by: $(u_i T_{ij}) - (u_j T_{ji})$

This representation uses basis vectors ($\mathbf{e}_i$) and dyadics ($\mathbf{e}_i \otimes \mathbf{e}_j$) to break the computation down into fundamental components.

Would you like further details on the steps or the interpretation of the result? Here are some related questions to explore further:

1. How does the antisymmetric part of a tensor influence the result of this operation?
2. How can we verify this computation in terms of physical or geometric properties?
3. What happens if the tensor $\mathbf{T}$ is symmetric?
4. How would this operation change in different coordinate systems?
5. Can this expression be used to define a skew-symmetric tensor?

Tip: Pay close attention to the indices' arrangement when working with tensors; they directly control the transformation properties and the resulting operations.