2.5 Tensor Gradient

Question 1

What is the covariant derivative equal to?

Answer 1

𝛿_mu = (1/c 𝛿/𝛿t, ∇)

Question 2

What is the contravariant derivative equal to?

Answer 2

𝛿^mu = (1/c 𝛿/𝛿t, -∇)

Question 3

What is the scalar invariant of the derivative operator equal to?

Answer 3

The d’Alembertian oeprator

Question 4

What is the four-current equal to?

Answer 4

(ρc, j) = charge density*c, three current vector

Question 5

What are Maxwell’s Equations in terms of the faraday tensor and the derivative?

Answer 5

𝛿_alpha F^(alpha beta) = mu_0 j^(beta)

Question 6

Which Maxwell equations can be expressed in terms of the derivative and the Faraday tensor?

Answer 6

The inhomogeneous ones i.e. the ones with a source

Question 7

What is the contravariant vector potential equal to?

Answer 7

A^(mu) = (Φ/c , A)

Question 8

What is the covariant vector potential equal to?

Answer 8

A^(mu) = (Φ/c , -A)

Question 9

How can the electric field be expressed in terms of the scalar and vector potentials?

Answer 9

E = -∇Φ - dA/dt

Question 10

What is the covariant Faraday tensor equal to in terms of the vector potential and derivative?

Answer 10

F_(mu nu) = 𝛿_mu A_nu - 𝛿_nu A_mu