Index Notation 2

Question 1

what is the curl of u, [∇ x u]i in index notation

Answer 1

• recall [a x b]i = ε_ijka_jb_k
• [∇ x u]i = ε_ijk∇_ju_k
• = ε_ijk*du_k/dx_j

Question 2

for positions vectors x, what does dx_i/dx_j equal

Answer 2

• dx_i/dx_j = δ_ij

Question 3

what is the divergence of a position vector in index notation, ∇ . x

Answer 3

• ∇ . x = dx_i/dx_i (partials) =δ_ii = 3
• because ∇ . x = dx_i/dx_i + dx_j/dx_j + dx_k/dx_k = 1 + 1 + 1 = 3

Question 4

what is the curl of a position vector in index notation, ∇ cross x

Answer 4

• [∇ cross x] = ε_ijkdx_k/dx_j = ε_ijkδ_jk = 0
• because of two repeated indices

Question 5

what is ∇|x|i in index notation

Answer 5

• ∇|x|i = (x_k*x_k)^-1/2 * x_i

Question 6

what is the divergence theorem

Answer 6

• int[∇.f dV] = int[f.n dS]

Question 7

what is stokes theorem

Answer 7

• int[∇ x f.dA] = int[f.dl]
• dA = n dS

Question 8

how can you know if it’s stokes or the divergence theorem being used in index notation

Answer 8

• ε on one side = stokes theorem
• ε on both sides = divergence theorem

Question 9

what is the directional derivative of a function f(u) defined as

Answer 9

• Df(u)[v] = d/dε * f(u + εv) | ε=0
• derivative of f with respect to u in the direction of v

Question 10

what would the directional derivative of f = u . u be

Answer 10

• Df(u)[v] = d/dε ((u + εv) . (u + εv)) | ε = 0
• = d/dε ((u + εv)^2) | ε = 0
• = 2(u + εv) . v | ε = 0 {chain rule}
  = 2u . v