Differential Area Element and surface integrals

In summary, When performing surface integrals, the correct differential area element to use is dA = dxdy. The equation A=xy does not make sense and to use the product rule, you must differentiate A with respect to either x or y. However, when performing line, surface, and volume integrals, it is always best to start with the infinitesimal element. Thank you to all who helped clarify this issue.
  • #1
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[SOLVED] Differential Area Element

While doing surface integrals, I am not sure as to which of the following is the correct differential area element to be considered:

i] [tex]dA = dx dy[/tex]

or

ii]
[tex]A = xy[/tex]

hence, using the product rule:

[tex]dA = xdy + ydx[/tex]
 
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  • #2
The short answer is simply that [tex]dA = dxdy[/tex]

Your second equation does not make sense. To use the product rule, you have to differentiate A with respect to either x or y.
 
  • #3
the first one ... you always start with the infinitesimal element .

CrazyIvan said:
The short answer is simply that [tex]dA = dxdy[/tex]

Your second equation does not make sense. To use the product rule, you have to differentiate A with respect to either x or y.

that's not completely correct ..
if you start with [tex]A=xy[/tex] then [tex]dA=xdy+ydx[/tex] is lgeit .. but we always start with the infinitesimal element when we perform line , surface and volume integrals .
 
Last edited:
  • #4
thanks for the quick replies. I think I get it now...
 

1. What is a differential area element?

A differential area element is a small area on a surface that is used to represent the surface in a surface integral. It is typically denoted as dA and is a product of two differential lengths, dx and dy.

2. How is a differential area element related to surface integrals?

A surface integral is the mathematical representation of calculating the area of a curved surface. The differential area element, dA, is used in the integration process to break the surface into small, manageable pieces and sum them up to find the total area.

3. Can a differential area element be represented in different coordinate systems?

Yes, a differential area element can be represented in different coordinate systems such as Cartesian, polar, or spherical. The choice of coordinate system depends on the shape and nature of the surface being integrated.

4. How do you calculate a surface integral using a differential area element?

To calculate a surface integral, you first need to choose an appropriate coordinate system and define the limits of integration. Then, you need to express the surface element (dA) in terms of the chosen coordinates. Finally, integrate the function over the limits defined to find the total area.

5. What is the significance of differential area element in physics and engineering?

Differential area elements are essential in physics and engineering as they allow for the calculation of surface integrals, which have various applications in these fields. For example, surface integrals are used to calculate the flux of a vector field across a surface, which is crucial in understanding fluid flow and electromagnetic fields.

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