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

  • Thread starter gikiian
  • Start date
  • #1
98
0
And what is the justification to consider or not to consider dy=dx?

-An Engineer, Weak in Calculus
 

Answers and Replies

  • #2
35,437
11,856
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.
 
  • #3
UVW
24
0
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.
 
  • #4
Redbelly98
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
12,122
160
Moderator's note: thread moved from Classical Physics to Calculus

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.

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.
 
  • #5
98
0
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?
 
  • #6
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
4,528
572
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.
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
41,833
964
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..
 
  • #8
98
0
dy can be greater than dx, dx can be greater than dy, or they could be equal. The are independent..

Thanks.
 

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

  • Last Post
Replies
8
Views
11K
Replies
20
Views
4K
Replies
5
Views
2K
Replies
12
Views
3K
Replies
12
Views
1K
  • Last Post
Replies
4
Views
2K
Replies
4
Views
3K
  • Last Post
Replies
4
Views
4K
  • Last Post
Replies
22
Views
12K
  • Last Post
Replies
5
Views
9K
Top