Coordinate systems/Derivation of formulas/Differential normal areas

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Why this formula for surface area element seems either wrong or misleading[edit | edit source]

The question[edit | edit source]

The question is whether it is appropriate to write the vector surface element as the sum of three perpendicular elements.

Answer[edit | edit source]

Changing notation a bit, we work in Cartesian coordinaes:

(1)  

is not how the w:surface integral is usually set up. Instead one uses a cross product and a surface defined in parametric form:

(2)  

Nevertheless, it is possible to force as shown in (1) to a vector area element with the following constraints:

(3)  

(4)  

In this context, writing (1) is putting the proverbial round peg into a square hole.[1] This can be seen by if we examine both differentials:

(5)     ,   and  

The paths and must follow the contour of the surface like a net. But one path cannot move in the z direction, while the other is similarly confined to a plane at constant x. Hence we can form a area differential of the form given by (1), but it cannot be integrated over any surface not confined to a flat plane. Hence, we conclude that (1) describes a differential surface element that cannot be integrated over any curved surface. Ultimately, there is no strong reason to replace (1) by

(6)  

Equation (6) has the advantage of not misleading the reader that the differential can be integrated over a surface, while (1) has the advantage of allowing the surface to be a flat plane at any orientation.

  1. The requirement that a differential be negative simple means that the sign of the differential and the component of the area element in the associated direction are opposite.