4. Mathematical operations involving tensors

Try these first

1. 1.

   What is the definition of a second-order tensor?

   Hint

   Think about how we visualize data in rows and columns.

2. 2.

   What is the Kronecker delta function?

   Hint

   It's a function that tells if two indices are equal.

3. 3.

   What is the result of multiplying a second-order tensor with a vector?

   • Another tensor
   • A scalar
   • A vector

   Hint

   Think about how we transform vectors using matrices.

4. 4.

   The Kronecker delta function equals 1 when the indices are...

   • True
   • False

   Hint

   Recall its defining property.

5. 5.

   Given a tensor A with components A_11 = 1, A_12 = 2, A_21 = 3, A_22 = 4 and a vector b = [5, 6], compute the product a = A*b and explain each step.

   Hint

   Follow matrix multiplication rules.

6. 6.

   Prove that multiplying two second-order tensors A and B gives a tensor C, and show how to derive C_ij from matrices A and B in standard form.

   Hint

   Apply the summation convention correctly.