Chapter 1.6, Problem 1.62P

(a)

To determine

The area vector of a hemispherical bowl of radius R .

(b)

To determine

To show: The area vector is equal to zero for any closed surface.

(c)

To determine

To show: The area, vector ais same for all the surfaces sharing the same boundary.

(d)

To determine

To show: The area of cone is given by $\overrightarrow{a}=\frac{1}{2}{\displaystyle \oint \overrightarrow{r}\times d\overrightarrow{l}}$ where integral is over boundary line of cone.

(e)

To determine

To prove: ${\displaystyle \oint (\overrightarrow{c}.\overrightarrow{r})d\overrightarrow{l}=\overrightarrow{a}\times \overrightarrow{c}}$ . For any constant vector c.Vector r is a position vector.