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Over the past years in math, I've seen many different methods for reparametrizing variables as a way to make a complex integral easier to deal with. Essentially we just take a function of some value and differentiate it with respect to a different variable, creating newer, simpler functions. One application of this includes u-substitution, where we let an arbitrary variable (in this case "u") represent a portion of our integral. For example, let's take this integral: | Over the past years in math, I've seen many different methods for reparametrizing variables as a way to make a complex integral easier to deal with. Essentially we just take a function of some value and differentiate it with respect to a different variable, creating newer, simpler functions. One application of this includes u-substitution, where we let an arbitrary variable (in this case "u") represent a portion of our integral. For example, let's take this integral: | ||
<center><math> \int {(sin{(x)})*(cos{(x)})} dx</math></center> | <center><math> \int {(sin{(x)})*(cos{(x)})} dx</math></center> | ||
| − | To solve this, we would simply let our new variable "u" equal <math>sin{(x)}</math> and differentiate both sides, resulting in an equation with <math> du = cos{(x)} dx </math>. We can then proceed to use this as a substitution for dx, changing our integral to <math> \int {sin{(u)} du}</math> | + | To solve this, we would simply let our new variable "u" equal <math>sin{(x)}</math> and differentiate both sides, resulting in an equation with <math> du = cos{(x)} dx </math>. We can then proceed to use this as a substitution for dx, changing our integral to <math> \int {sin{(u)} du}</math>, which is much easier to compute. |
| − | , | + | While u-substitution is the clearest example of parametrization, we even see it appear in surface integrals. This concept is extremely useful, especially with complex integrals, and it plays a major role in an integration technique known as Feynman's technique. |
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[[ Walther MA271 Fall2020 topic14 | Back to Feynman Integrals]] | [[ Walther MA271 Fall2020 topic14 | Back to Feynman Integrals]] | ||
Revision as of 16:35, 27 November 2020
A Review on Reparametrizing
Over the past years in math, I've seen many different methods for reparametrizing variables as a way to make a complex integral easier to deal with. Essentially we just take a function of some value and differentiate it with respect to a different variable, creating newer, simpler functions. One application of this includes u-substitution, where we let an arbitrary variable (in this case "u") represent a portion of our integral. For example, let's take this integral:
To solve this, we would simply let our new variable "u" equal $ sin{(x)} $ and differentiate both sides, resulting in an equation with $ du = cos{(x)} dx $. We can then proceed to use this as a substitution for dx, changing our integral to $ \int {sin{(u)} du} $, which is much easier to compute.
While u-substitution is the clearest example of parametrization, we even see it appear in surface integrals. This concept is extremely useful, especially with complex integrals, and it plays a major role in an integration technique known as Feynman's technique.
