Integrating Vector Area

Integrating vector expressions over areas is much different than integrating scalar expressions. This is illustrated nicely by the difference between a scalar area and a so-called vector area.

QUESTIONS
(Score is number right minus number wrong.)

On a hemisphere, the correct SCALAR differential area in spherical coordinates is

: (1a); (1b); (1c); (1d)

On a hemisphere, the correct VECTOR differential area in spherical coordinates is

: Equation (2a); Equation (2b); Equation (2c); Equation (2d)

The SCALAR area of the surface of a hemisphere or radius R is correctly given by Equation (3).

: True; False

The VECTOR area of the surface of a hemisphere or radius R is correctly given by

: Equation (4a); Equation (4b); Equation (4c); Equation (4d)

Which is FALSE?

: Equation (5a); Equation (5b); Equation (5c)

For purposes of integration over a hemisphere, Cartesian unit vectors are constant, but spherical unit vectors are not constant.

: True; False

The VECTOR area of the surface of a hemisphere or radius R is correctly given by Equation (6).

: True; False

Which is TRUE?

: Equation (7a); Equation (7b); Equation (7c)

EQUATIONS

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