Differentials
- Thread starter Oxymoros
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Simple answer: Yes and No.
Those symbols serve various purposes. In the expression [math]\int f(x)\;dx[/math] the "dx" is just a reminder of what variable we are using. The expression would be the same without the "dx", so long as the intent was clear.
The symbol [math]\dfrac{dy}{dx}[/math] represents the result of a limiting process defined by a ratio. However, if we are talking about it's finite approximation, we might consider it a division. Of course, this also happens in differential equations where we are talking about the limiting process.
No question that it can be a little confusion.
Those symbols serve various purposes. In the expression [math]\int f(x)\;dx[/math] the "dx" is just a reminder of what variable we are using. The expression would be the same without the "dx", so long as the intent was clear.
The symbol [math]\dfrac{dy}{dx}[/math] represents the result of a limiting process defined by a ratio. However, if we are talking about it's finite approximation, we might consider it a division. Of course, this also happens in differential equations where we are talking about the limiting process.
No question that it can be a little confusion.
D
Deleted member 4993
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Those are operators (d/dx) - but act like multiplication and division as long as you are dealing with function of single variable [like y = f(x)]
I have read people calling it a "memory" of division and multiplication because they are the standard parts of the division Δy/Δx and multiplication Δx*Δy ,where Δy , Δx infinitesimal numbers (numbers that are the smaller than every positive real and greater than every negative real number, but not zero) . The standard parts dy/dx and dxdy, are the real numbers
,Δy/Δx and Δx*Δy are infinitively close to. Δy/Δx is infinitively close to dy/dx etc.
pka
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If you find the last of your sentences interesting there is available a free download a calculus text book. It is Elementary Calculus: An Infinitesimal Approach by Jerome Keisler. The chapters and whole book is a free down-load at HERE
Chapter one is on your question.
yoscar04
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Regarding [MATH]\int[/MATH]f(x)dx you can go back to the Riemann definition of the integral as [MATH]\sum[/MATH]f(xi)[MATH]\Delta[/MATH]xi. It has to do with the integral representing the area under a function f. The area under the curve is divided into a large number of rectangles with high f(xi) and width [MATH]\Delta[/MATH]xi. The area of each rectangle is the multiplication of the value of the function at a particular point (height of the rectangle) times the width of the rectangle. The area will be the sum of all the areas of all the rectangles under the curve y=f(x). I believe that reading about the Riemann integral will help you understand more about the notation.