Hold a small piece of paper (e. g., an index card) flat in front of you. the paper can be thought of as a part of a larger plane surface.

a. what single line could you use to specify the orientation of the plane of the paper (i. e., so that someone else could hold the paper in the same, or in a parallel, plane)?
b. the area of a flat surface can be represented by a single vector, called the area vector a. what does the direction of the vector represent? what would you expect the magnitude of the vector to represent?
c. place a large piece of graph paper flat on the table. describe the direction and magnitude of the area vector, a, for the entire sheet of paper. describe the direction and magnitude of the area vector, da, for each of the individual squares that make up the sheet.
d. fold the graph paper twice so that it forms a hollow triangular tube. can the entire sheet be represented by a single vector with the characteristics you defined above? if not, what is the minimum number of area vectors required?
e. form the graph paper into a tube as shown. can the orientation of each of the individual squares that make up the sheet of graph paper still be represented by da vectors as inabove? explain.
f. what must be true about a surface or a portion of a surface in order to be able to associate a single area vector a with that surface?

Explanation:

(a). The line used to specify the orientation of the plane of paper is the line normal to the plane of sheet of paper

(b). The direction of the vector represents the normal to the  lat surface while the Magnitude represents the area of flat surface.

(c). Say the area of each smaller square is 1 square unit, then the area of graph paper is 64 square units. Direction of this area vector is given by a unit vector perpendicular to the graph sheet. If X and Y axes are in the plane of paper, then unit vector normal to the sheet of paper is K. Hence the complete vector is 64 K sq. units.

Area vector of each individual square is 1 squ. unit. where all these individual squares are parallel as vectors.

(d). Absolutely.

the entire sheet can be represented by a single vector. Its area vector is the sum of area vectors of three flat sides of triangular tube.

(e) NO.

Orientation of the individual squares is not the same for all squares. They cannot be represented by the same vector when compared to part C above, because they are in different directions even tough their magnitude are same.

(f) To represent a surface with a single area vector, divide the surface in to as many as possible flat pieces (if necessary infinitely large number of infinitesimally small pieces). Find the area vectors of all pieces. Add all the area vectors to obtain the single area vector resenting the complete surface.

But since the process can be done for any surface, any surface can be represented by a single area vector.

i hope this helps, cheers