Let vx x2 x3e1 x3 x1e2 X1 x2e3 Determine the expressio

Let v(x) = (x_2 - x_3)e_1 + (x_3 - x_1)e_2 + (X_1 - x_2)e_3. Determine the expressions for nabla(x middot v) and nabla middot [x v], where x = x_i e_i is the position vector, and then evaluate the expressions at the point x = e_1 + 2e_2 + e_3. If a vector a = e_1 + 2e_2 + 3e_3 and the components of a symmetric second-order tensor T are: [T]_ij ~ [1 -1 -2 -1 2 -3 -2 -3 1], then evaluate the following: T_ii, T_ij T_ij and epsilon_ijk T_ij a_k.

Solution

o evaluate and manipulate tensors, we express them as components in a basis, just as for vectors. We can use the displacement gradient to illustrate how this is done. Let u(x1,x2,x3) be a vector field, and let G=u represent the gradient of u.  Recall the definition of G

du=Gdx

Now, let {e1,e2,e3} be a Cartesian basis, and express both du and dx as components. Then, calculate the components of du in terms of dx using the usual rules of calculus

From this example we see that G can be represented as a 3×3 matrix. The elements of the matrix are known as thecomponents of G in the basis {e1,e2,e3}.

The difference between a matrix and a tensor

If a tensor is a matrix, why is a matrix not the same thing as a tensor? Well, although you can multiply the three components of a vector u by any 3×3 matrix,

b1b2b3=a11a21a31a12a22a32a13a23a33u1u2u3

the resulting three numbers (b1,b2,b3) may or may not represent the components of a vector. If they are the components of a vector, then the matrix represents the components of a tensor A, if not, then the matrix is just an ordinary old matrix.

To check whether (b1,b2,b3) are the components of a vector, you need to check how (b1,b2,b3) change due to a change of basis. That is to say, choose a new basis, calculate the new components of u in this basis, and calculate the new matrix in this basis (the new elements of the matrix will depend on how the matrix was defined. The elements may or may not change – if they don’t, then the matrix cannot be the components of a tensor). Then, evaluate the matrix product to find a new left hand side, say (1,2,3). If  (1,2,3) are related to (b1,b2,b3) by the same transformation that was used to calculate the new components of u, then (b1,b2,b3) are the components of a vector, and, therefore, the matrix represents the components of a tensor.