I'm not sure. It looks overwhelmingly complicated to me.Orodruin said:
You might want to consider the start and end coordinates of the path (corresponding to ##t=0## and ##t= \pi##). You might want to consider whether or not the field is conservative.Graham87 said:
Compare the first terms of each expression. What do you see?Graham87 said:
Looks like some derivative or primitive function variation. But it's not in a series.Orodruin said:
There is really no "magic" too it. Just that ##3 x^2 y## is quite clearly the ##x##-derivative of ##x^3 y## ... of which ##x^3## is the ##y##-derivative, etc etcGraham87 said:
A line integral solution for curve γ is a mathematical concept that involves integrating a function along a curve in a vector field. It represents the total effect of the vector field along the curve.
Simplifying substitutions help in solving line integrals for curve γ by making the integration process easier and more manageable. By substituting variables or simplifying expressions, the integral can be solved more efficiently.
Some common simplifying substitutions used in solving line integrals for curve γ include trigonometric substitutions, algebraic substitutions, and parametric substitutions. These substitutions help simplify the integrand and make the integration process more straightforward.
Determining the appropriate simplifying substitution to use for a specific line integral for curve γ often involves analyzing the integrand and identifying patterns or structures that can be simplified. Practice and experience can also help in recognizing which substitution will be most effective.
Some tips for effectively using simplifying substitutions in solving line integrals for curve γ include practicing different substitution techniques, understanding the properties of the integrand, and being familiar with common substitution patterns. It is also helpful to break down the integral into smaller parts to simplify the overall calculation.