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

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

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

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

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

A contravariant vector transforms as:

A covariant vector transforms as:

What distinguishes a tensor from an arbitrary indexed array?

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

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

For \(T^i{}_j=\begin{pmatrix}1&4\\2&-1\end{pmatrix}\), compute \(T^i{}_i\).

For \(A_{ij}=\begin{pmatrix}3&1\\5&2\end{pmatrix}\), the off-diagonal value \(S_{12}\) of the symmetric part is:

The metric tensor \(g_{ij}\) is of type:

In plane polar coordinates \(ds^2=dr^2+r^2d\theta^2\), the metric matrix is:

The inverse metric for \(ds^2=dr^2+r^2d\theta^2\) is:

In polar coordinates, if \(v^r=2,\ v^\theta=-3\), then \(v_\theta\) equals:

For the polar metric, \(\Gamma^r{}_{\theta\theta}\) is:

For the polar metric, \(\Gamma^\theta{}_{r\theta}\) is:

The covariant derivative of a contravariant vector is:

In polar coordinates, let \(V^r=r,\ V^\theta=0\). Compute \(\nabla_\theta V^\theta\).

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

The geodesic equation is:

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

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

The scalar Laplacian in plane polar coordinates is: