A Particular Integrals of PDEs

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The discussion focuses on understanding the steps for specific integrations highlighted in a page from "Mathematical Methods for Physics and Engineering" by Riley, Hobson, and Bence. It emphasizes that if a function's derivative is zero, it must be a constant, and for functions of two variables, a zero derivative with respect to one variable indicates dependence solely on the other variable. Participants suggest exploring functions with zero mixed second partial derivatives to deepen understanding. Engaging with counterexamples is recommended as a method to clarify concepts. Overall, the conversation aims to clarify the principles behind particular integrals in partial differential equations.
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Partial integration
Can someone please explain the steps for the integrations in red circles on the attached page? This a page from Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 2006 edition.
Thank you.
 

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vgarg said:
TL;DR Summary: Partial integration

Can someone please explain the steps for the integrations in red circles on the attached page? This a page from Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 2006 edition.
Thank you.
If you have a function of a single variable whose derivative is zero, then that function must be a constant. If you have a function of two variables whose derivative is zero with respect to one of the variables, then it must be a function only of the other variable.
 
PS you should try to find a function of two variables whose mixed second partial derivative is zero. In general, I'd you don't understand something like this, trying to find a counterexample or otherwise prove it's wrong is is good way to understand why it's correct.
 
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