Area element in rectangular cartesian co-ordinate system:

Let us consider a parallelogram represented by the vectors \( \vec{P} \) and \( \vec{Q} \) as shown in the adjoining figure 1. So the area vector of the parallelogram is given by,

\( Area =\vec{P}\times\vec{Q} \), since the area is a vector product of two vectors, so the area is a vector quantity and the direction of this area vector is normal to the plane containing the two vectors \( \vec{P} \) and \( \vec{Q} \) and it can be determined by the rule of the cross-product of two vectors.

Let us consider a rectangle ABCD in a cartesian co-ordinate, lying in the X-Y plane, as shown in figure 2.

The co-ordinate of the point \( A \) with respect to the origin \( O \) is \( (x,y) \),

The co-ordinate of the point \( B \) with respect to the origin \( O \) is \( (x+dx,y) \),

The co-ordinate of the point \( C \) with respect to the origin \( O \) is \( (x+dx,y+dy) \),

The co-ordinate of the point \( D \) with respect to the origin \( O \) is \( (x,y+dy) \)

Therefore, \( \vec{AB}=(\vec{x}+\vec{dx})-\vec{x}=\vec{dx}=dx\ \hat{i} \)

and, \( \vec{AD}=(\vec{y}+\vec{dy})-\vec{y}=\vec{dy}=dy\ \hat{j} \)

where, \( \hat{i} \) and \( \hat{j} \) are the two unit vectors along the X-axis and Y-axis respectively.

So the area of ABCD is given by, \( \vec{dA}=\vec{dx}\times\vec{dy}=dx\hat{i}\times{dy}\hat{j}=dx\ dy\ \hat{k} \)

where, \( \hat{k} \) is the unit vector along the Z-axis.