Solving Some Tensor Problems: Confirm Grad Dot (y x sigma) = y x Grad Dot Sigma

[ponjavic]

Homework Statement

1a) Confirm that grad dot (y x sigma) = y x grad dot sigma + grad*sigma x sigma
where y is a vector and sigma a second order tensor

Homework Equations

(a x b)_i = e_ipq a_p b_q
(A x B)_i = e_ipq A_pl B_ql
(a x B)_ij = e_ipq a_p B_qj

(caps are tensors)

The Attempt at a Solution

I get:
e_ipq y_p sigma_qj = e_ipq y_p sigma_qj,j + e_ipq y_p,l sigma_ql

if it is correct I'm not quite sure how to add the third and second term together

 

[mighty2000]

.

Dear forum post author,

Thank you for your question regarding the confirmation of the equation grad dot (y x sigma) = y x grad dot sigma + grad*sigma x sigma. it is important to ensure the accuracy and validity of equations, so I am happy to assist you in verifying this equation.

Firstly, I would like to clarify that grad dot (y x sigma) is a scalar quantity, while y x grad dot sigma and grad*sigma x sigma are both second order tensors. Therefore, the equation we are confirming is actually:

grad dot (y x sigma) = y x grad dot sigma + grad*sigma x sigma

To solve this equation, we can start by writing out the components of each term using the given equations for vector and tensor products. We get:

grad dot (y x sigma) = (e_ipq y_p sigma_qj),j

y x grad dot sigma = e_ipq y_p (sigma_qj),j

grad*sigma x sigma = e_ipq (sigma_pl),j (sigma_qj)

Next, we can use the product rule for differentiation to expand the first term on the right-hand side:

(e_ipq y_p sigma_qj),j = (e_ipq y_p,j sigma_qj) + (e_ipq y_p sigma_qj,j)

= (e_ipq y_p,l sigma_qj) + (e_ipq y_p sigma_qj,j)

= (e_ipq y_p,l sigma_qj) + (e_ipq y_p,l sigma_qj) + (e_ipq y_p,l sigma_qj)

= 2(e_ipq y_p,l sigma_qj) + (e_ipq y_p sigma_qj,j)

We can see that the first and second terms on the right-hand side match with the third and fourth terms on the left-hand side, respectively. This means that the equation is indeed confirmed.

I hope this helps to clarify the steps and reasoning behind the solution. Please feel free to ask for any further clarification or assistance.