Vector Analysis and Coordinate Systems Questions and Answers

Answer: C

The magnitude of the cross product is the area, and for perpendicular vectors it is simply the product of the magnitudes. Halving it would give the area of the triangle the two vectors span, which is a related standard result.

Answer: D

It behaves as a vector in that it may be dotted or crossed with a field or multiplied into a scalar, but it consists of differentiations and so has no value on its own. That dual character is why gradient, divergence and curl are all written with the same symbol.

Answer: A

For perpendicular vectors the resultant is the square root of the sum of the squares, and \( \sqrt{25 + 144} = 13 \). The arithmetic sum of seventeen would require them to be parallel and the difference of seven antiparallel.

Answer: B

The metric factor appears both inside and outside the differentiation, which is what distinguishes the curvilinear form from the Cartesian one. Expanding it gives the second derivative plus a first derivative divided by the radius.

Answer: C

Holding the radial distance fixed gives a cylinder about the axis, holding the azimuthal angle fixed gives a half plane containing the axis, and holding the height fixed gives a plane. The three intersect at right angles, which is what makes the system orthogonal.

Answer: D

A fixed radius gives a sphere about the origin, a fixed polar angle gives a cone with its apex there, and a fixed azimuthal angle gives a half plane containing the axis. Their mutual perpendicularity is again what makes the system usable.

Answer: A

Matching the coordinate system to the symmetry reduces the number of variables the field depends on, often from three to one, and that saving far outweighs the extra complexity of the curvilinear operators. The transformation itself is a routine substitution.

Answer: B

For a force field it is literally work; for the electric field it is the potential difference with the sign reversed. Only when the path is closed does it become a circulation, and for a conservative field that closed value is zero.

Answer: C

Correct answer: \( \int \mathbf{A} \cdot d\mathbf{s} \)

Flux counts only the component of the field passing through the surface, so the dot product with the vector area element is exactly right: it takes the normal component and discards the tangential one. Using the magnitude alone would count field lines running along the surface, which pass through nothing.

Answer: D

The direction cosines are the components of the unit vector along the given direction, and a unit vector has magnitude one, so the sum of the squares of its components is one. This is a useful check on any set of direction cosines computed.