Question

10. For the given Tensor T, (a) Decompose the tensor into a symmetric and an antisymmetric part, (b) Find the dual vector for the antisymmetric part, (c) Verify $\mathbf{T}^A \cdot \vec{a} = \vec{t}^A \times \vec{a}$ for $\vec{a} = \hat{e}_1 + \hat{e}_3$. Note: $^A$ represents antisymmetric, and $\vec{t}^A$ is the dual vector of antisymmetric tensor.

          10. For the given Tensor T, (a) Decompose the tensor into a symmetric and an antisymmetric part, (b) Find the dual vector for the antisymmetric part, (c) Verify $\mathbf{T}^A \cdot \vec{a} = \vec{t}^A \times \vec{a}$ for $\vec{a} = \hat{e}_1 + \hat{e}_3$.
Note: $^A$ represents antisymmetric, and $\vec{t}^A$ is the dual vector of antisymmetric tensor.

10. For the given Tensor T, (a) Decompose the tensor into a symmetric and an antisymmetric part, (b) Find the dual vector for the antisymmetric part, (c) Verify 𝐓^A ·a⃗ = t⃗^A ×a⃗ for a⃗ = ê1 + ê3.
Note: ^A represents antisymmetric, and t⃗^A is the dual vector of antisymmetric tensor.

Added by Steven G.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Madhur L

10/13/2022

Step 1

To decompose a tensor into its symmetric and antisymmetric parts, we can use the following formulas: Symmetric part: T^(S) = 1/2 * (T + T^T) Antisymmetric part: T^(A) = 1/2 * (T - T^T) where T^T is the transpose of tensor T. Show more…

Show all steps

Thanks for your feedback!

For the given Tensor T, (a) Decompose the tensor into a sym- metric and an antisymmetric part, (b) Find the dual vector for the antisymmetric part, (c) Verify T^(A)*vec(a)=vec(t)^(A) imes vec(a) for vec(a)=hat(e)_(1)+hat(e)_(3). Note: ^(A) represents antisymmetric, and vec(t)^(A) is the dual vector of anti- symmetric tensor. 10. For the given Tensor T, (a) Decompose the tensor into a sym- metric and an antisymmetric part, (b) Find the dual vector for the Note: A represents antisymmetric, and tA is the dual vector of anti- symmetric tensor.