Topic 2: Tensor Calculus

Question 1

Scalar Field

Answer 1

A field that associates a scalar to every point in a space.

It is a real valued function of the position vector.

Question 2

Vector Field

Answer 2

A field that assigns a vector to every point in a space.

It is a vector whose individual components are given by scalar fields.

Question 3

Tensor Field

Answer 3

A field that assigns a tensor to every point in a space.

It is a tensor whose components are given by scalar fields.

Question 4

Express G = Del outerproduct of v (where v is a vector) in index notation.

Answer 4

G_ij = v_j,i

Question 5

Express H = v (where v is a vector) outerproduct Del in index notation.

Answer 5

G_ij = v_i,j

Question 6

Divergence of a vector field v

Answer 6

Div of v = Del dot v

V_i,i

Question 7

Divergence of a tensor field T

Answer 7

Div of T = T dot Del

T_ij,j

Question 8

Laplacian of T

Answer 8

Del dot gradient of T

Question 9

Curl of v

Answer 9

Curl of v = Del cross v

t_i = eps_ijk v_k,j

Question 10

Divergence theorem formula

Answer 10

The surface integral of Q_ijk…q n_q dA = the volume integral of (Q_ijk…q),q dV

Question 11

partial(xi) / partial(xj) = ?

Answer 11

delta_ij