Vector Analysis and Coordinate Systems Questions and Answers

Answer: A

It is a vector whose direction is the normal to the loop giving the greatest circulation, fixed by the right hand rule. Applied to the magnetic field it gives the current density, which is Ampere's law in differential form.

Answer: B

Correct answer: the divergence of the gradient of that field

\[ \nabla^2 V = \nabla\cdot(\nabla V). \] The gradient turns the scalar into a vector and the divergence turns it back into a scalar. The other combinations offered are either identically zero - the curl of a gradient and the divergence of a curl both vanish - or are not defined for a scalar at all.

Answer: C

Correct answer: \( \partial A_x/\partial x + \partial A_y/\partial y + \partial A_z/\partial z \)

Each component is differentiated with respect to its own coordinate and the three are added, giving a scalar. The first option is one component of the curl, which is built from the cross derivatives, and the second is the Laplacian of a vector.

Answer: D

A gradient field has no circulation, so its curl vanishes identically. This is why a field that can be written as the gradient of a potential is necessarily irrotational, and it is the basis of the electrostatic potential.

Answer: A

A curl field has no sources, so its divergence vanishes identically. This is why the magnetic flux density, being the curl of the vector potential, satisfies \( \nabla \cdot \mathbf{B} = 0 \) and magnetic monopoles do not appear.

Answer: B

Correct answer: solenoidal

A solenoidal field has no sources or sinks, so its flux lines close on themselves and the flux out of any closed surface is zero. The magnetic flux density is the standard example, and it is why magnetic monopoles do not appear in Maxwell’s equations.

Answer: C

Correct answer: irrotational or conservative

No circulation means the curl vanishes and every closed line integral is zero, so the integral between two points depends only on the endpoints. That is what allows a potential function to be defined, and it is why the electrostatic field can be written as the gradient of a voltage.

Answer: D

Correct answer: a volume integral of the divergence to a surface integral of the flux

\[ \int_V (\nabla\cdot\mathbf{A})\,dv = \oint_S \mathbf{A}\cdot d\mathbf{s}. \] It converts a statement about sources throughout a volume into one about flux across its boundary, and it is what connects the differential and integral forms of Gauss’s law.

Answer: A

Correct answer: a surface integral of the curl to a line integral round the bounding curve

\[ \int_S (\nabla\times\mathbf{A})\cdot d\mathbf{s} = \oint_C \mathbf{A}\cdot d\mathbf{l}. \] It is the two-dimensional counterpart of the divergence theorem, converting circulation over a surface into circulation round its edge - and it is what connects the two forms of Ampere’s law.

Answer: B

Correct answer: zero

A conservative field is the gradient of a scalar, so the line integral between two points is just the difference of that scalar at the ends. Round a closed path the ends coincide and the difference is zero. This is why moving a charge round any closed path in an electrostatic field requires no net work.