Index Notation

Question 1

Define the Kronecker delta.

Answer 1

Question 2

Show how the identity matrix I is linked with the Kronecker delta.

Answer 2

Question 3

What is the following in index notation.

Answer 3

Question 4

What is the symbol for the Levi-Civita?

Answer 4

ε_ijk

Question 5

What does ε_ijk stand for?

Answer 5

The Levi-Civita

Question 6

Define ε_ijk, the Levi-Civita by two of it’s properties.

Answer 6

Question 7

What is another name for this property?

Answer 7

Antistymmetric

Question 8

What five other properties does these inital two properites of ε_ijk imply?

Answer 8

Question 9

How do you write the vector cross product C = A x B in index notation?

Answer 9

Question 10

When is something called a dummy indice?

Answer 10

When it is summed over in the formula and so don’t appear in the final answer

Question 11

What does the following equal?

Answer 11

Question 12

Prove the following be equating both sides

Answer 12

Question 13

What is the Einstein Summation Convention?

Answer 13

You drop the summation sign, with the convention that wherever an index is repeated you sum over it.

Question 14

Finish the following equations using Einstein’s summation convention.

Answer 14

Question 15

What is the difference between dummy and free indices?

Answer 15

Dummy indices are indices to be summed other where as free will remain in the answer

Question 16

What is a generic rule you can use for δ in equations like the following.

Answer 16

If δ has a dummy indice then delete the δ and replace the dummy index in the rest of the expression by the other index on the deleted δ.

Question 17

What is the i^th component of _∇_f in index notation?

Answer 17

Question 18

What is the i^th component in ∇.v?

Answer 18

Question 19

What is the i^th component of ∇ x v?

Answer 19

Question 20

What does the following equal?

Answer 20