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Differentiation is a function that takes a function and spits out a different one. (This may fail at points, but wherever the function is smooth enough, you get

by joppy 5y ago

Differentiation is a function that takes a function and spits out a different one. (This may fail at points, but wherever the function is smooth enough, you get a corresponding point on the derivative). It is relatively simple to define, if somewhat laborious to make precise (real analysis etc). The most straightforward definition of indefinite integration on the other hand is “the inverse of differentiation”. Like a lot of functions, inverses are much more complicated. Even in the case of matrix-vector multiplication, the inverse function (the inverse matrix) has an expression in terms of determinants in the original entries of the matrix - much more complicated than the original problem. So I think that it’s the more simple theme of: even if the function is easy to describe, its inverse may be extremely complicated.

Kranar 5y ago
You could always say integration is a function that takes a function and spits out another one, and differentiation is the inverse of integration. There's no intrinsic reason why an inverse is anymore difficult or why the derivative is the "base" and integration is the inverse. The more abstract reason differentiation is easier than integration is that because differentiation is a local operation, given a family of functions differentiation will be closed within that family. That is the derivative of an elementary function is an elementary function, the derivative of a rational function is a rational function, the derivative of a function f is a function f' that is within the same family of f and will share almost all of the same properties as f. Integration on the other hand does not have this property, and in fact it's very difficult to find family of functions that are closed under integration. The integral of a rational function may not be rational, the integral of an elementary function may not be elementary. Integration is a global operation whose solution may require the construction of a new family of functions whose properties are entirely unlike the function being integrated. It's like how squaring a number is usually closed within that number system, but taking the square root of a number could require creating entirely new number systems, whether it's real numbers or complex numbers (at which point the square root is closed). With integration, the issue is that it never ends up being closed, you can use integration on a family of functions to jump out of that family and construct entirely new families of functions seemingly without end. In a way, differentiation produces functions whose properties are a subset of the original function, but integration produces functions whose properties expand upon the original function.

by joppy 5y ago

Kranar 5y ago

You could always say integration is a function that takes a function and spits out another one, and differentiation is the inverse of integration. There's no intrinsic reason why an inverse is anymore difficult or why the derivative is the "base" and integration is the inverse. The more abstract reason differentiation is easier than integration is that because differentiation is a local operation, given a family of functions differentiation will be closed within that family. That is the derivative of an elementary function is an elementary function, the derivative of a rational function is a rational function, the derivative of a function f is a function f' that is within the same family of f and will share almost all of the same properties as f. Integration on the other hand does not have this property, and in fact it's very difficult to find family of functions that are closed under integration. The integral of a rational function may not be rational, the integral of an elementary function may not be elementary. Integration is a global operation whose solution may require the construction of a new family of functions whose properties are entirely unlike the function being integrated. It's like how squaring a number is usually closed within that number system, but taking the square root of a number could require creating entirely new number systems, whether it's real numbers or complex numbers (at which point the square root is closed). With integration, the issue is that it never ends up being closed, you can use integration on a family of functions to jump out of that family and construct entirely new families of functions seemingly without end. In a way, differentiation produces functions whose properties are a subset of the original function, but integration produces functions whose properties expand upon the original function.