Question 1

In $A^i{}_jB^j{}_k$, the resulting tensor type is:

Accepted Answer

(1,1)

Answer

(0,2)

Answer

(0,0)

Answer

(2,0)

Question 2

The quantity $a_i v^i$ is invariant because:

Accepted Answer

the Jacobian and inverse Jacobian cancel

Answer

both are covariant

Answer

it has two free indices

Answer

both are contravariant

Question 3

In the general tensor transformation law, each lower index contributes:

Accepted Answer

an inverse Jacobian factor

Answer

a Christoffel symbol

Answer

a determinant only

Answer

a Jacobian factor

Question 4

For $u=2x+y,\ v=x-3y$, the Jacobian $ \frac{\partial(u,v)}{\partial(x,y)} $ is:

Accepted Answer

-7

Answer

7

Answer

-5

Answer

5

Question 5

Under $u=2x+y,\ v=x-3y$, a contravariant vector $(v^x,v^y)=(1,-2)$ becomes:

Accepted Answer

(0,7)

Answer

(0,-5)

Answer

(7,0)

Answer

(4,-5)

Question 6

Under $u=x+y,\ v=x-y$, a covariant vector $(a_x,a_y)=(1,5)$ becomes:

Accepted Answer

(3,-2)

Answer

(6,-4)

Answer

(-2,3)

Answer

(2,3)

Question 7

If $A^{ij}$ is type $(2,0)$ and $B_j$ is type $(0,1)$, then $A^{ij}B_j$ is of type:

Accepted Answer

(1,0)

Answer

(2,1)

Answer

(0,1)

Answer

(1,1)

Question 8

For $T^i{}_j=\begin{pmatrix}2&-1\4&3\end{pmatrix}$, the contraction $T^i{}_i$ is:

Accepted Answer

5

Answer

3

Answer

-1

Answer

6

Question 9

In $R^{ij}{}_{jk}$, after contracting over $j$, the resulting type is:

Accepted Answer

(1,1)

Answer

(0,2)

Answer

(2,2)

Answer

(2,0)

Question 10

The quotient rule says: if $A_i v^i$ is a scalar for every contravariant vector $v^i$, then $A_i$ is:

Accepted Answer

a covariant vector

Answer

a scalar

Answer

a contravariant vector

Answer

not a tensor

Question 11

For $A_{ij}=\begin{pmatrix}2&8\0&4\end{pmatrix}$, the skew-symmetric component $K_{12}$ is:

Accepted Answer

4

Answer

8

Answer

2

Answer

0

Question 12

Christoffel symbols are not tensors because:

Accepted Answer

their transformation law contains extra derivative terms

Answer

they have no indices

Answer

they are always zero

Answer

they are always symmetric in all three indices

Question 13

The covariant derivative of a covariant vector is:

Accepted Answer

$\nabla_ja_i=\partial_ja_i-\Gamma^k{}_{ij}a_k$

Answer

$\nabla_ja_i=\Gamma^i{}_{jk}a^k$

Answer

$\nabla_ja_i=\partial_i a_j$

Answer

$\nabla_ja_i=\partial_ja_i+\Gamma^k{}_{ij}a_k$

Question 14

In polar coordinates, if $W^r=0,\ W^\theta=r$, then $\nabla_r W^\theta$ is:

Accepted Answer

2

Answer

r

Answer

1

Answer

0

Question 15

In polar coordinates, if $V^r=0,\ V^\theta=1$, then $\nabla_\theta V^r$ is:

Accepted Answer

$-r$

Answer

$r$

Answer

0

Answer

$1/r$

Question 16

The $\theta$-geodesic equation in plane polar coordinates is:

Accepted Answer

$\ddot\theta+\frac{2}{r}\dot r\dot\theta=0$

Answer

$\ddot\theta+r\dot r^2=0$

Answer

$\ddot\theta-r\dot\theta^2=0$

Answer

$\ddot r+\frac{2}{r}\dot r\dot\theta=0$

Question 17

The Riemann curvature tensor measures:

Accepted Answer

failure of covariant derivatives to commute

Answer

ordinary matrix determinant

Answer

the trace of the metric

Answer

only scale factors

Question 18

If all components of $R^i{}_{jkl}$ vanish in a region, then the region is:

Accepted Answer

locally flat

Answer

not differentiable

Answer

certainly singular

Answer

without any coordinates

Question 19

Why can the polar plane have nonzero Christoffel symbols but zero curvature?

Accepted Answer

Because Christoffel symbols reflect coordinate-basis change

Answer

Because polar coordinates are impossible

Answer

Because the metric is singular everywhere

Answer

Because tensors do not transform

Question 20

The Ricci tensor is obtained from the Riemann tensor by:

Accepted Answer

$R_{jl}=R^i{}_{jil}$

Answer

$R_{jl}=R^i{}_{ijl}$

Answer

$R_{jl}=\partial_j\partial_lg$

Answer

$R_{jl}=g_{jl}R$

Question 21

The scalar curvature is:

Accepted Answer

$R=g^{jl}R_{jl}$

Answer

$R=\delta^i{}_i$

Answer

$R=g_{jl}R^{jl}$ only if no inverse metric exists

Answer

$R=\partial_iA^i$

Question 22

In the tensor expression for curl, the Christoffel part cancels mainly because:

Accepted Answer

epsilon is antisymmetric and Gamma is symmetric in lower indices

Answer

epsilon is symmetric and Gamma is antisymmetric

Answer

the metric is zero

Answer

both are scalars

Question 23

The divergence formula in cylindrical coordinates using physical components is:

Accepted Answer

$\frac1r\partial_r(rA^{(r)})+\frac1r\partial_\theta A^{(\theta)}+\partial_zA^{(z)}$

Answer

$\partial_rA^{(r)}+\partial_\theta A^{(\theta)}+\partial_zA^{(z)}$

Answer

$\frac1{r^2}\partial_r(rA^{(r)})+\partial_zA^{(z)}$

Answer

$r\partial_rA^{(r)}+\partial_\theta A^{(\theta)}+\partial_zA^{(z)}$

Question 24

The polar Laplacian of a scalar $\phi(r,\theta)$ is:

Accepted Answer

$\frac1r\partial_r(r\partial_r\phi)+\frac1{r^2}\partial_{\theta\theta}\phi$

Answer

$\partial_{rr}\phi+\partial_{\theta\theta}\phi$

Answer

$\frac1{r^2}\partial_r\phi+\partial_{\theta\theta}\phi$

Answer

$\partial_r(r\phi)+\partial_\theta\phi$

Question 25

For $\phi(r,\theta)=r^3$, the polar Laplacian $\Delta\phi$ is:

Accepted Answer

$9r$

Answer

$0$

Answer

$6r$

Answer

$3r$

Related quizzes

Related quizzes