Vector Analysis and Coordinate Systems Questions and Answers

Answer: C

Measured from the positive z axis, the polar angle reaches \( \pi \) at the negative z axis and that is the whole of its range; the azimuthal angle then sweeps the full \( 2\pi \) round. Allowing \( \theta \) to exceed \( \pi \) would cover space twice.

Answer: D

Correct answer: \( \rho \, d\rho \, d\phi \, dz \)

The three edges of the element are \( d\rho \), \( \rho\,d\phi \) and \( dz \), the middle one carrying a factor of \( \rho \) because an angle subtends a longer arc further from the axis. Multiplying them gives the volume element, and omitting the \( \rho \) is the commonest error in cylindrical integration.

Answer: A

The two angular arc lengths are \( r \, d\theta \) and \( r \sin\theta \, d\phi \), and multiplying them by \( dr \) gives the element. Integrating it over the full ranges returns \( 4\pi r^3/3 \), which is the check worth remembering.

Answer: B

Each component must be a length, so each angular differential is multiplied by the radius of the circle it sweeps. For the azimuthal angle that radius is \( r \sin\theta \), the distance from the z axis rather than from the origin.

Answer: C

The radial unit vector points away from the axis or origin, so it points differently at different places. This is why they cannot simply be moved outside a derivative or an integral, and why gradient, divergence and curl take different forms in each system.

Answer: D

Correct answer: \( \mathbf{a}_z \)

The unit vectors form a right-handed set in the order \( \rho, \phi, z \), so each crossed into the next gives the third: \( \mathbf{a}_\rho\times\mathbf{a}_\phi = \mathbf{a}_z \), \( \mathbf{a}_\phi\times\mathbf{a}_z = \mathbf{a}_\rho \), and \( \mathbf{a}_z\times\mathbf{a}_\rho = \mathbf{a}_\phi \). Reversing the order reverses the sign.

Answer: A

Correct answer: \( \rho \cos\phi \)

Since \( \phi \) is measured from the x axis, the projection onto that axis is \( \rho\cos\phi \) and onto the y axis \( \rho\sin\phi \). The z coordinate is common to both systems, so the conversion touches only the two transverse coordinates.

Answer: B

The polar angle is measured from the z axis, so the projection of the radius onto that axis is the cosine. The projection onto the xy plane is \( r \sin\theta \), which then divides between x and y by the azimuthal angle.

Answer: C

Correct answer: in the direction of the greatest rate of increase of the scalar

The gradient points uphill, along the steepest ascent, and its magnitude is the rate of change in that direction. It is therefore perpendicular to the surfaces on which the scalar is constant - which is why the electric field, being the negative gradient of potential, is perpendicular to the equipotentials.

Answer: D

Correct answer: the net outward flux per unit volume from a vanishing volume there

Divergence is a source density: it reports how much of the field originates at the point, as the flux out of a small volume divided by that volume in the limit. That is exactly what makes \( \nabla\cdot\mathbf{D} = \rho_v \) the local form of Gauss’s law.