Question 1

In Einstein notation, an index repeated exactly twice, once upstairs and once downstairs, means:

Accepted Answer

sum over that index

Answer

differentiate with respect to that index

Answer

set the index equal to zero

Answer

take absolute value

Question 2

In T^i{}_j v^j, the resulting object is:

Accepted Answer

a contravariant vector

Answer

a scalar

Answer

a covariant vector

Answer

a rank-2 tensor

Question 3

Which expression is invalid as a pure tensor contraction unless a metric is used first?

Accepted Answer

u_i v_i

Answer

delta^i{}_j v^j

Answer

T^i{}_i

Answer

a_i v^i

Question 4

The Kronecker delta satisfies delta^i{}_j v^j =

Accepted Answer

v^i

Answer

delta^i{}_i

Answer

0

Answer

v_i

Question 5

For u=x+y, v=x-y, the Jacobian  ∂(u,v)/∂(x,y)  is:

Accepted Answer

-2

Answer

2

Answer

0

Answer

1/2

Question 6

A contravariant vector transforms as:

Accepted Answer

bar v^i=∂ bar x^i/∂ x^jv^j

Answer

bar v_i=∂ x^i/∂ bar x^jv^j

Answer

bar v_i=∂ bar x^i/∂ x^jv_j

Answer

bar v^i=∂ x^j/∂ bar x^iv^j

Question 7

A covariant vector transforms as:

Accepted Answer

bar a_i=∂ x^j/∂ bar x^ia_j

Answer

bar a_i=∂ bar x^i/∂ x^ja_j

Answer

bar a_i=a_i in every coordinate system

Answer

bar a^i=∂ x^i/∂ bar x^ja_j

Question 8

What distinguishes a tensor from an arbitrary indexed array?

Accepted Answer

its transformation law

Answer

whether it has a determinant

Answer

whether its entries are integers

Answer

number of components only

Question 9

If u^i is contravariant and a_j is covariant, T^i{}_j=u^i a_j is of type:

Accepted Answer

(1,1)

Answer

(0,0)

Answer

(2,0)

Answer

(0,2)

Question 10

If T^i{}_{jk} is contracted over i and j, the result T^i{}_{ik} is of type:

Accepted Answer

(0,1)

Answer

(1,0)

Answer

(0,0)

Answer

(1,1)

Question 11

For T^i{}_j=begin{pmatrix}1&42&-1end{pmatrix}, compute T^i{}_i.

Accepted Answer

0

Answer

3

Answer

1

Answer

-3

Question 12

For A_{ij}=begin{pmatrix}3&15&2end{pmatrix}, the off-diagonal value S_{12} of the symmetric part is:

Accepted Answer

3

Answer

5

Answer

2

Answer

1

Question 13

The metric tensor g_{ij} is of type:

Accepted Answer

(0,2)

Answer

(1,1)

Answer

(2,0)

Answer

(0,0)

Question 14

In plane polar coordinates ds^2=dr^2+r^2dtheta^2, the metric matrix is:

Accepted Answer

mathrm{diag}(1,r^2)

Answer

mathrm{diag}(1,r)

Answer

mathrm{diag}(1,1/r^2)

Answer

mathrm{diag}(r^2,1)

Question 15

The inverse metric for ds^2=dr^2+r^2dtheta^2 is:

Accepted Answer

mathrm{diag}(1,1/r^2)

Answer

mathrm{diag}(r^2,1)

Answer

mathrm{diag}(1/r,1/r)

Answer

mathrm{diag}(1,r^2)

Question 16

In polar coordinates, if v^r=2, v^theta=-3, then v_theta equals:

Accepted Answer

-3r^2

Answer

-3r

Answer

-3

Answer

-2r^2

Question 17

For the polar metric, Gamma^r{}_{thetatheta} is:

Accepted Answer

-r

Answer

1/r

Answer

0

Answer

r

Question 18

For the polar metric, Gamma^theta{}_{rtheta} is:

Accepted Answer

1/r

Answer

r

Answer

0

Answer

-r

Question 19

The covariant derivative of a contravariant vector is:

Accepted Answer

nabla_j v^i=∂_jv^i+Gamma^i{}_{jk}v^k

Answer

nabla_j v^i=∂_jv^i-Gamma^i{}_{jk}v^k

Answer

nabla_j v^i=Gamma^k{}_{ij}v_k

Answer

nabla_j v^i=∂_i v^j

Question 20

In polar coordinates, let V^r=r, V^theta=0. Compute nabla_theta V^theta.

Accepted Answer

1

Answer

0

Answer

r

Answer

1/r

Question 21

Parallel transport of V^i(t) along a curve is defined by:

Accepted Answer

DV^i/dt=0

Answer

V^i=0

Answer

Gamma^i{}_{jk}=0

Answer

∂_i V^i=0

Question 22

The geodesic equation is:

Accepted Answer

ddot x^i+Gamma^i{}_{jk}dot x^jdot x^k=0

Answer

∂_i g_{jk}=0

Answer

dot x^i+Gamma^i{}_{jk}x^jx^k=0

Answer

nabla_i A^i=0

Question 23

For the Euclidean plane in polar coordinates, R^r{}_{theta rtheta} is:

Accepted Answer

0

Answer

r^2

Answer

-1

Answer

1/r

Question 24

For a contravariant vector field A^i, the compact divergence formula is:

Accepted Answer

nabla_iA^i=1/sqrt g∂_i(sqrt g A^i)

Answer

nabla_iA^i=sqrt g,∂_i A^i

Answer

nabla_iA^i=epsilon^{ijk}∂_jA_k

Answer

nabla_iA^i=g_{ij}A^j

Question 25

The scalar Laplacian in plane polar coordinates is:

Accepted Answer

Deltaphi=frac1r∂_r(r∂_rphi)+frac1{r^2}∂_{thetatheta}phi

Answer

Deltaphi=∂_{rr}phi+∂_{thetatheta}phi

Answer

Deltaphi=frac1{r^2}∂_r(r∂_rphi)+∂_{thetatheta}phi

Answer

Deltaphi=r∂_rphi+∂_{thetatheta}phi

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