(a) A robotic arm's sensor sits at the point m, giving plane polar radius m. Write…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Tensors in flat spacetime: index notation and covariant field equations

Problem

(a) A robotic arm's sensor sits at the point

(x,y) = (8, 6)

m, giving plane polar radius

r = 10

m. Write down

\partial r/\partial x

\partial r/\partial y

\partial\phi/\partial x

\partial\phi/\partial y

at this point, then find the polar components

(dr,d\phi)

produced by the Cartesian displacement

dx^\mu = (0, 0.020\ \mathrm{m}, 0.010\ \mathrm{m}, 0)

. (b) A four-vector has Cartesian components

A^\mu = (5, 3, 0, 0)

m. Lower the index and contract to find

A^\mu A_\mu

. State what type of object results and whether the interval is timelike, spacelike or null. (c) A two-index array has

Q^{02} = 4

and

Q^{20} = -2

(all other components zero). Find

Q^{(02)}

and

Q^{[02]}

. (d) The fields at an event are

\vec E = (0, 2.0\times10^5, 0)

V m

^{-1}

and

\vec B = (5.0\times10^{-4}, 0, 0)

T. Give

F_{02}

and

F_{23}

(take

c=3.0\times10^8

m s

^{-1}

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