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

• ponjavic
In summary, a tensor is a mathematical object used to describe relationships between different coordinate systems. Solving tensor problems is important in various fields of science and engineering, and it involves confirming the transformation rules and solving complex calculations. Grad Dot (y x sigma) is a mathematical operation that represents the directional derivative of y along the direction of sigma. Confirming the equality of Grad Dot (y x sigma) and y x Grad Dot Sigma is important for verifying calculations and ensuring correct transformation rules are followed. This problem can be solved by using tensor properties and breaking it down into smaller steps.
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

.

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:

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.

## 1. What is a tensor?

A tensor is a mathematical object that describes the relationship between different coordinate systems in a multi-dimensional space. It is represented by a set of components and follows specific transformation rules.

## 2. What is the significance of solving tensor problems?

Solving tensor problems is important in various fields of science and engineering, such as physics, mathematics, and computer science. It helps us understand the relationships between different physical quantities and simplifies complex calculations.

## 3. What is Grad Dot (y x sigma)?

Grad Dot (y x sigma) is a mathematical operation that involves taking the gradient of the vector product of two tensors, y and sigma. In other words, it represents the directional derivative of y along the direction of sigma.

## 4. Why is it important to confirm Grad Dot (y x sigma) = y x Grad Dot Sigma?

Confirming this equality ensures that the transformation rules for tensors are being followed correctly. It also helps in verifying the correctness of calculations and identifying any errors in the process.

## 5. How can we solve the tensor problem "Confirm Grad Dot (y x sigma) = y x Grad Dot Sigma"?

This problem can be solved by using the properties of tensor operations and applying the appropriate transformation rules. It is also helpful to break down the problem into smaller steps and double-check each step for accuracy.

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