When we take a differential area dxdy do we assume that dy=dx?

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Discussion Overview

The discussion revolves around the assumptions related to differential area elements in calculus, specifically whether dy can be considered equal to dx. Participants explore the implications of this assumption in various contexts, including curves and areas, and the justification for or against it.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants question the validity of assuming dy = dx, suggesting that it is not a general rule and can only apply in specific cases.
  • One participant illustrates the relationship between dy and dx using the example of the function y = sin(x), indicating that dy and dx depend on the curve and the specific point on it.
  • Another participant notes that while differential areas are often visualized as squares for ease of integration, there is no inherent reason for this assumption, especially when considering numerical techniques.
  • There is a discussion about the relationship between dy and dx in the context of an ellipse, with one participant suggesting that dy could be greater than, less than, or equal to dx, emphasizing their independence.

Areas of Agreement / Disagreement

Participants express differing views on the assumption of dy = dx, with no consensus reached. Some argue against the assumption while others provide specific contexts where it might apply.

Contextual Notes

The discussion highlights the lack of a general justification for assuming dy = dx and the dependence of this relationship on the specific problem or context being considered.

gikiian
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And what is the justification to consider or not to consider dy=dx?

-An Engineer, Weak in Calculus
 
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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

gikiian said:
And what is the justification to consider or not to consider dy=dx?
It's not assumed because (in general) there is no justification in assuming it. Pretty much as mfb said.

UVW said:
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.
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.
 
UVW said:
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.

Redbelly98 said:
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 f(x,y) = \cos(x+100y)
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.
 
gikiian said:
If we take a differential area inside an ellipse with major axis along y, then will dy be greater than dx?
dy can be greater than dx, dx can be greater than dy, or they could be equal. The are independent..
 
HallsofIvy said:
dy can be greater than dx, dx can be greater than dy, or they could be equal. The are independent..

Thanks.
 

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