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Mathematics
Calculus
Why is this definite integral a single number?
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[QUOTE="fresh_42, post: 6900618, member: 572553"] It is an analytical concept and the definitions require limits. For a purely algebraic point of view, you would first have to establish such an environment so that the terms make sense. You can consider a differentiation as a derivation (no typo!) or a differential operator, but the corresponding integral operator uses an integral in its definition, i.e. again an analytical term. The infinitesimal notation - in this context - is more of a reminder not to forget that we deal with limits (change of) rather than having a meaning of their own. $$\left. \dfrac{d}{d\,x}\right|_{x_0}f(x)=\left. \dfrac{d\,f}{d\,x}\right|_{x_0} =\displaystyle{\lim_{h \to 0}}\dfrac{f(x_0+h)-f(x_0)}{h}.$$ You cannot discuss them separately. Best you can get is Weierstrass's formula $$ f(x_0+h)=f(x_0)+ \underbrace{\left(\left. \dfrac{d}{d\,x}\right|_{x_0}f(x)\right)}_{=D_{x_0}f}\cdot h + r(h) $$ where all the infinitesimal quantities, all limits, anything going to zero, all those are put into the correction term ##r(h)## which is required to converge faster to zero than ##h## does, and ##D_{x_0}f## is merely a linear function and not the quotient of infinitesimals. In other contexts, especially in physics, ##dx## can also mean a basis vector of the cotangent space or a differential, a Pfaffian form. These cases do not use quotients of ##d## expressions and can thus be explained without limits, or better: with hidden limits since we still use tangents. That was more of a rhetorical statement. Meteorology is full of (tangent) vector fields. You see at least one whenever you see a weather forecast, often literally with drawn vectors. Yet, the way from a bunch of secants converging to a tangent up to the theorem of the hedgehog as we call the hairy ball theorem is long, or to meteorology. Here (pretty much at the beginning) is a list of 10 perspectives on the differentiation process: [URL]https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/[/URL] and I didn't even use the word [I]slope[/I]. You have to go this way if you study physics, at least the first steps to cotangent spaces and differential forms. If you want to study physics on a deeper level, then you will probably need pushforwards and pullbacks, too. [/QUOTE]
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Mathematics
Calculus
Why is this definite integral a single number?
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