Graduate Calculating Functional Derivatives: -1≤xₒ≤1 vs -1<xₒ<1

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SUMMARY

The discussion focuses on the calculation of functional derivatives, specifically comparing the cases of -1 ≤ xₒ ≤ 1 and -1 < xₒ < 1. The integral representation of the functional derivative is given as δI[f]/δf(xₒ) = ∫_a^b δ(x - xₒ) dx with limits a = -1 and b = +1. The participants debate the implications of the boundary conditions on integration by parts, concluding that the choice of boundary conditions is significant for the correctness of the integration process.

PREREQUISITES
  • Understanding of functional derivatives
  • Familiarity with the Dirac delta function
  • Knowledge of integration by parts
  • Basic concepts of calculus and boundary conditions
NEXT STEPS
  • Study the properties of the Dirac delta function in functional analysis
  • Explore advanced integration techniques, particularly integration by parts
  • Investigate the implications of boundary conditions in calculus
  • Learn about functional derivatives in the context of variational calculus
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Mathematicians, physicists, and students engaged in advanced calculus, particularly those working with functional analysis and variational principles.

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A functional derivative using the delta function with intergral limits of +-1.
##\frac {\delta I[f]} {\delta f(x_o)} = \int_a ^b \delta(x-x_o) \, dx## with a=-1 and b=+1

## -1 \leq x_o \leq +1 ## vs ## -1 \lt x_o \lt +1 ##, 0 otherwise. Which is correct and does it matter when doing integration by parts?
 
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