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An entire subfield of analysis called Stenciling exists, for this purpose. Depending on the function or differential equation / system, different stencils are u

An entire subfield of analysis called Stenciling exists, for this purpose. Depending on the function or differential equation / system, different stencils are used. Stenciling doesn't just deal with the derivative but it tries to come up with approximations involving a fixed number of sample points, the stencil, for any differential operator, ie. the Laplacian, higher order derivatives, etc. > The term "stencil" was coined for such patterns to reflect the concept of laying out a stencil in the usual sense over a computational grid to reveal just the numbers needed at a particular step. [2] [0] https://en.wikipedia.org/wiki/Five-point_stencil https://en.wikipedia.org/wiki/Five-point_stencil [1] https://en.m.wikipedia.org/wiki/Finite_difference_coefficient#Arbitrary_stencil_points https://en.m.wikipedia.org/wiki/Finite_difference_coefficien... [2] https://en.wikipedia.org/wiki/Stencil_(numerical_analysis) https://en.wikipedia.org/wiki/Stencil_(numerical_analysis)

yarg 4y ago
Thanks for the links (and the flame war). Another thing that I'm wondering about is how to calculate the length of a function. What I'm thinking, is that you can treat a function like a piece of string, you take f(x) and generate a pair of functions, one above and one below f(x). You generate the functions at a fixed (infinitesimal) tangential distance 𝛿t and generate deltas for {x, y} mul = 𝛿t/((𝛿x)^2 + (𝛿y)^2)^.5 Δx = 𝛿y * mul Δy = 𝛿x * mul u(x - Δx) = f(x) + Δy l(x + Δx) = f(x) - Δy The idea being that the function length is the area ∫(u(x) - l(x)) divided by the thickness 2 * 𝛿t. I have no idea how wrong this is, but if you can point me toward any resources that'll help when I (inevitably) get stuck, it would be very much appreciated.
- goldenkey 4y ago
  Not exactly sure what you are getting at but it does remind me of the Cauchy's Integral Formula which shows that differentiation is equivalent to integration. https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula
  - yarg 4y ago
    Trying to measure the length of a curve f(x). I turn the curve into a fixed infinitesimal thickness (2 * 𝛿t) 'rope' and measure the area. Then I divide the area by the thickness to get the length. The rope is defined as the area between two curves, both of them manipulations of f(x). The curves are generated by moving each point (in f(x)) a constant infinitesimal distance at a tangent to the function (this gives the rope its fixed thickness), one in either direction. For the upper curve {Δx, Δy} = {(n * 𝛿y), -(n * 𝛿x)}, for the lower {(n * 𝛿y), -(n * 𝛿x)}, where n is a normalisation constant 𝛿t/((𝛿x)^2 + (𝛿y)^2)^.5 Once you have these two functions, you can use the integral of the difference to get the area. That's what I'm thinking anyway.
    - goldenkey 4y ago
      I'm a brainlet or maybe it's the 10+ hours of work I did today but I just can't grasp it right now. There are a lot of ways to take integrals though, so I wouldn't doubt your method could work. Generally though the holes in a lot of these integration methods is their inability to work with functions that are pathological: https://en.wikipedia.org/wiki/Pathological_(mathematics) https://en.wikipedia.org/wiki/Pathological_(mathematics) |group1 = Types of integrals |list1 = * [[Riemann integral]] * [[Lebesgue integration|Lebesgue integral]] * [[Burkill integral]] * [[Bochner integral]] * [[Daniell integral]] * [[Darboux integral]] * [[Henstock–Kurzweil integral]] * [[Haar measure|Haar integral]] * [[Hellinger integral]] * [[Khinchin integral]] * [[Kolmogorov integral]] * [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes integral]] * [[Pettis integral]] * [[Pfeffer integral]] * [[Riemann–Stieltjes integral]] * [[Regulated integral]] |group2 = Integration techniques |list2 = * [[Integration by substitution|Substitution]] * [[Trigonometric substitution|Trigonometric]] * [[Euler substitution|Euler]] * [[Weierstrass substitution|Weierstrass]] * [[Integration by parts|By parts]] * [[Integration by partial fractions|Partial fractions]] * [[Integration using Euler's formula|Euler's formula]] * [[Integral of inverse functions|Inverse functions]] * [[Order of integration (calculus)|Changing order]] * [[Integration by reduction formulae|Reduction formulas]] * [[Integration using parametric derivatives|Parametric derivatives]] * [[Leibniz integral rule#Evaluating definite integrals|Differentiation under the integral sign]] * [[Laplace transform#Evaluating improper integrals|Laplace transform]] * [[Contour integration]] * [[Laplace's method]] * [[Numerical integration]] * [[Simpson's rule]] * [[Trapezoidal rule]] * [[Risch algorithm]]

yarg 4y ago

Thanks for the links (and the flame war). Another thing that I'm wondering about is how to calculate the length of a function. What I'm thinking, is that you can treat a function like a piece of string, you take f(x) and generate a pair of functions, one above and one below f(x). You generate the functions at a fixed (infinitesimal) tangential distance 𝛿t and generate deltas for {x, y} mul = 𝛿t/((𝛿x)^2 + (𝛿y)^2)^.5 Δx = 𝛿y * mul Δy = 𝛿x * mul u(x - Δx) = f(x) + Δy l(x + Δx) = f(x) - Δy The idea being that the function length is the area ∫(u(x) - l(x)) divided by the thickness 2 * 𝛿t. I have no idea how wrong this is, but if you can point me toward any resources that'll help when I (inevitably) get stuck, it would be very much appreciated.

goldenkey 4y ago
Not exactly sure what you are getting at but it does remind me of the Cauchy's Integral Formula which shows that differentiation is equivalent to integration. https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula
- yarg 4y ago
  Trying to measure the length of a curve f(x). I turn the curve into a fixed infinitesimal thickness (2 * 𝛿t) 'rope' and measure the area. Then I divide the area by the thickness to get the length. The rope is defined as the area between two curves, both of them manipulations of f(x). The curves are generated by moving each point (in f(x)) a constant infinitesimal distance at a tangent to the function (this gives the rope its fixed thickness), one in either direction. For the upper curve {Δx, Δy} = {(n * 𝛿y), -(n * 𝛿x)}, for the lower {(n * 𝛿y), -(n * 𝛿x)}, where n is a normalisation constant 𝛿t/((𝛿x)^2 + (𝛿y)^2)^.5 Once you have these two functions, you can use the integral of the difference to get the area. That's what I'm thinking anyway.
  - goldenkey 4y ago
    I'm a brainlet or maybe it's the 10+ hours of work I did today but I just can't grasp it right now. There are a lot of ways to take integrals though, so I wouldn't doubt your method could work. Generally though the holes in a lot of these integration methods is their inability to work with functions that are pathological: https://en.wikipedia.org/wiki/Pathological_(mathematics) https://en.wikipedia.org/wiki/Pathological_(mathematics) |group1 = Types of integrals |list1 = * [[Riemann integral]] * [[Lebesgue integration|Lebesgue integral]] * [[Burkill integral]] * [[Bochner integral]] * [[Daniell integral]] * [[Darboux integral]] * [[Henstock–Kurzweil integral]] * [[Haar measure|Haar integral]] * [[Hellinger integral]] * [[Khinchin integral]] * [[Kolmogorov integral]] * [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes integral]] * [[Pettis integral]] * [[Pfeffer integral]] * [[Riemann–Stieltjes integral]] * [[Regulated integral]] |group2 = Integration techniques |list2 = * [[Integration by substitution|Substitution]] * [[Trigonometric substitution|Trigonometric]] * [[Euler substitution|Euler]] * [[Weierstrass substitution|Weierstrass]] * [[Integration by parts|By parts]] * [[Integration by partial fractions|Partial fractions]] * [[Integration using Euler's formula|Euler's formula]] * [[Integral of inverse functions|Inverse functions]] * [[Order of integration (calculus)|Changing order]] * [[Integration by reduction formulae|Reduction formulas]] * [[Integration using parametric derivatives|Parametric derivatives]] * [[Leibniz integral rule#Evaluating definite integrals|Differentiation under the integral sign]] * [[Laplace transform#Evaluating improper integrals|Laplace transform]] * [[Contour integration]] * [[Laplace's method]] * [[Numerical integration]] * [[Simpson's rule]] * [[Trapezoidal rule]] * [[Risch algorithm]]