/sci/ - Science & Math

>>6872677
Inverses usually are harder than doing something

Probably because you usually learn differentiation first and integration is Topsy-Turvy.

>>6872677
Why is multiplying easier than dividing?
Why is squaring easier than square rooting?
Why is exponentiating easier than taking logarithm?

Derivatives of elementary functions are elementary. Integrals of elementary functions are (in general) not.

Of course, "elementary function" s a completely arbitrary label, but we teach too it which means integrals end up looking alien in the context of prior knowledge.

Integration is easier because integral operators are usually continuous which isn't true for differential operators.

That is just the point of view of analytic maths and closed forms, in practical problems it is much more difficult to find numerically the derivative of something you don't actually know because it becomes buried in noise

>>6872682
>Inverses of inverses are much harder than doing something

In a heuristic way you could say differentiation usually creates complexity, whereas integration reduces complexity. Things that create complexity are easier to do.

I blame the rule of chains (and the jews).

>>6872677
BECAUSE : you have an unique solution to a derivate. But integration go througth primitive of the function you integer, and there's an infinite solution of primitive.

There are also function analyhticaly diffined as the integer of an other function, and some mathematician had mathematicaly proven that there is no analytical solution to describe this function other than describing it by an integer.

This is the MATHEMATICIAN vision.

The physicist vision is,

If you consider your function as a curve, then it is a 1 Dimension system. When you integer it, you calculate the aera of this curve in reference to you basis. So that mean it become a 2 Dimensions system.

Each time you integer you add 1 Dimension to the system, that mean you add complexity (since it enhance the degre of freedom)

>>6872677
because differentiation is just about applying (g°f)' recursively. Combining derivatives of elementary functions is just a computation problem.

Meanwhile integrals can't always be evaluated in closed form in terms of elementary functions
(http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra))

To piggyback off of the other posts, there are two main points of view: algebraic & analytic.

Algebraically, say you have the ring of polynomials. The derivative is closed under this ring, i.e. takes the ring of polynomials to the ring of polynomials. It's also closed under the elementary functions, and since all calculations by hand are pretty much done with elementary functions, it gives a sense that differentiation is easier. Integration (i.e., anti-differentiation) is *not* closed in the elementary functions. This is studied in "differential galois theory".

From an analytic point of view, the anon who said something about creating complexity is more or less correct, though the proper term is regularity, or degree of differentiability:

http://en.wikipedia.org/wiki/Smoothness

If you have a function f, Df is going to be "closer" to discontinuous than Int(f) the anti-derivative; so think of it as a "credit" system. The more you differentiate, the more you spend your "smoothness" credits, and you earn "smoothness credits" when you take the antiderivative. Why is it "easier" to make things discontinuous? Well, it depends on what you mean by easy, but in a vaguely signal-processing sense, you're losing information when you differentiate and gaining information when you integrate (really, take the anti-derivative).

>>6872726
Example:
|x| is continuous.
D(|x|) is discontinuous.
Int(|x|) is twice differentiable.
Int^n(|x|) is n+1-times differentiable.

>>6872726
>though the proper term is regularity, or degree of differentiability

sorry, french physicist here

For the high schooler differentiation is easier because you can easily get closed form solutions.
For the mathematican integrals are easier because they behave better under limits.
For the engineer integrals are easier because they can be easier approximated.

I don't understand why people dislike the error function so much but have no problems with other integral functions like the Gamma function.

They're equally difficult,
<div class="math">\widehat{\frac{\partial^n}{\partial x^n}f}(\xi) = (2 \pi i \xi)^n \hat{f}(\xi)</div>
hence,
<div class="math">\hat{f}(\xi) = \frac{\widehat{\frac{\partial^n}{\partial x^n}f}(\xi)}{(2 \pi i \xi)^n}</div>

>>6872794
Allow me a second try.

They're equally difficult,
<div class="math">\widehat{\frac{\partial^n}{\partial x^n}f}(\xi) = (2 \pi i \xi)^n \hat{f}(\xi)</div>
hence,
<div class="math">\hat{f}(\xi) = \frac{\widehat{\frac{\partial^n}{\partial x^n}f}(\xi)}{(2 \pi i \xi)^n}</div>

>>6872794
>>6872797
Third time's the charm?

>>6872794
you just wrote the derivation definition of fourrier transform, you're trying to scare childs ?

>>6872800
There is a newline inserted somewhere.

Part of buggy 4chan code.

Same thing happens if you spam a HTML entity like ", <, > or &.

I'll just enter the output from a sane Latex interpreter.

>>6872690
Why?

>>6872804
>There is a newline inserted somewhere.
The bottom latex line is too long, it's formatted as reading text before being passed to jsmath so it sticks a newline between the p and a in \partial.

>why is the differintegral so elementary at positive integer values
that's the oddball question, since at every other value the expression is fucked up

>>6872677
Taking derivatives is easier for a few reasons. For one, the methods to take derivatives of a given function are pretty explicitly defined. The other reason is because, when first learning about calculus, you work with elementary functions. Stuff like x^2, Sin[x], Exp[x], Ln[x]. Differentiating these functions is easy because they will yield more elementary functions. Integrating these functions can be difficult because they may yield non elementary functions such as your pic.