If you like, yes. A volume in mathematics is an oriented quantity. E.g. a length from left to right is say L, then it is -L from right to left. The volume of a parallelepiped spanned by vectors ##v_1,\ldots,v_n## is the determinate of these vectors, or the norm of ##v_1\wedge\ldots\wedge v_n##. Either way, if we swap two vectors, then the sign of the determinant as well as of the wedge product changes. This is also true for integrals.
That depends on what you call a Riemann sum. If you only consider the areas, then it is width times height and you have no orientation. But if you strictly follow the formulas, then height as well as length is oriented, simply because ##x_n-x_{n-1} \neq x_{n-1}-x_n## and the same for height.
But this would complicate the entire calculus. E.g. ##\int_{-1}^1 x^3\,dx = 0## oriented, which is why we calculate ##|\int_{-1}^0 x^3\,dx|+ |\int_{0}^1 x^3\,dx| = \frac{1}{2}## if we actually want to know the size of the area under the curve. Now if you do this on the level of Riemann sums, then you would get something like ##\lim_{n \to \infty} \sum \,\left|\dfrac{i-1}{n} -\dfrac{i}{n}\right|\cdot \left|\left(\dfrac{x_{i-1}-x_i}{2}\right)^3\right|## which is much more complicated, esp. if the example is more complicated. It is easier to work with oriented volumes and decide what to do if the actual purpose is clear: antiderivative or absolute volume?
I meant calculus in the sense of a framework for calculations, not as analysis. It is unfortunate that English doesn't distinguish calculus (frame) from calculus (analysis) or algebra (arithmetics, calculation) from algebra (theory of objects defined by one or two operations).
No. Algebra and calculus are both mathematics and they have touching points, but are basically two different things. They only share the property, that they are ambiguously used.
You're on the wrong side of history here. Solving an even simpler equation such as 2x - 3 = 1 would have been considered algebra by al-Khwarizmi, back in about 830 AD. For the next thousand years, algebra was synonymous with the idea of solving equations. I know that you conflate arithmetic and algebra (and we've had this conversation before), but in my view, arithmetic is exclusively associated with the so-called arithmetic operations -- addition, subtraction, multiplication, and division, without letters being used for variables. IOW, in my view, basic math and arithmetic are pretty much the same.
I would categorize this as Modern Algebra (i.e., concepts introduced only in the past 100 +-40 years.fresh_42 said:
My objection was to the conflation of algebra with "school math," or worse, with arithmetic. Leaving off discussion of "an algebra or an algebra over a field," which has a very precise meaning, I don't see the need to raise the confusion level by using a new term for something that's been around for a very long time. I'm not at all confused when the conversation is about algebra versus abstract/modern algebra.
Well, not really, since Diophantine equations are equations with integer solutions. At least here in the US, these kinds of equations are studied typically in a class on Number Theory, which is university-level.
We in the US also distinguish between calculus (we don't usually call it a calculus) and analysis, with analysis involving a lot of proofs with sequences, as well as measure theory and so on.fresh_42 said:
That's another example of the English penchant for substitution of a specific for something properly more general. As a literary trope it's metonymy. An educated man may be called a 'man of letters'. In the US, we sometimes say a person 'schools' someone, when he educates someone, including outside of the context of a school.fresh_42 said:
I was looking for the english counterpart of Kalkül. Google translates it with calculus, but there is no english Wikipedia version for Kalkül, and chrome translates it with calculus again. @WWGD told me - I think it was in General Discussions but I couldn't find it - that calculus is used in the sense of Kalkül in english, too, i.e. as a calculation machinery for a specific goal as in my examples in post #12. The Riemann calculus of integration is not the same as if I just say calculus. The former is only a small part of the latter; and the calculus of first order logic is something completely different than calculus.
Partly because Latin and Greek are 'perfect' languages, ('perfect' here being used the sense of meaning per-fect, or completed or 'past', as in the Perfect Tense of the Latin verb), incipiators of emergent disciplines have tended to prefer those ancient languages over in-flux modern languages when naming such disciplines or their intellectionary artifacts.
And dental hygienists remove calculus from your teeth, also in the "small stones" sense of the word.
Right. The root word is used rather promiscuously. A dental professional is more apt to refer to 'the calculus', in the sense of 'calculus' as 'stuff' rather than to 'a calculus' as a thing (or in the plural, to 'calculi' as things, as urologists might) , while a mathematician referring to 'the calculus' would most likely be referring to the marvelous methods invented by Newton and Leibniz, and decried by Berkely for the associated inconsistency in the treatment of an infinitesimal as non-zero when it is one of an infinitude of them being summed, and as zero when it is individuated and disregarded at the asymptotic limit.
Would you please explain if "oriented" here means "have a direction", such as "vectors" do?
Yes, volumes have a direction, but the usual term is orientation.
You are right, we only consider the positive value a length, area or volume.
A professor whose math class I took said something like "To see at once the beauty of the calculus, observe how it bridges between algebra and geometry". I liked that. His PhD was in Physics, but he was a family man, with the associated pressing obligations, and the job he was offered by the university was teaching math. He said he thought he'd have preferred to teach physics, but he was happy teaching math, in large part because it had so much that was so beautiful.