And what is the justification to consider or not to consider dy=dx?
-An Engineer, Weak in Calculus
Where did you see "dy=dx"? In general, this is wrong.
It is like "y=x". It can be true, in some specific problem, but it is meaningless as general equation.
mfb is correct. Let's consider the equation y = sin(x). Then dy/dx = cos(x). If we use the chain rule (or pretend that dy/dx is a fraction for a moment), we find that dy = cos(x)*dx.
So dy and dx are going to change their relationship depending on the curve (or plane, etc.) that we're considering, and also on where we are on the curve.
Moderator's note: thread moved from Classical Physics to Calculus
It's not assumed because (in general) there is no justification in assuming it. Pretty much as mfb said.
I don't think that addresses the OP's question, since they were asking about a differential area element rather than the slope of a curve.
If we take a differential area inside an ellipse with major axis along y, then will dy be greater than dx?
Pictorially and conceptually, you usually assume that your differential areas are squares, because there's no reason for them to have any other shape (and it especially makes doing iterative integration easier to visualize), but there's no specific reason for it to be true as long as you imagine some sort of partition of the plane whose elements are all going to zero in area.
For numerical techniques however it is often advantageous to not have squares. For example if I wanted to integrate [itex] f(x,y) = \cos(x+100y) [/itex]
The function is changing a lot more along the y direction than the x direction, so if I wanted to split up my region into rectangles, and take the value of f at the middle of the rectangle, multiply by the area and add them all up, I would be better off having my rectangles be a lot longer in the x direction than the y direction from an accuracy/time to calculate trade off.
dy can be greater than dx, dx can be greater than dy, or they could be equal. The are independent..
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