Greens theorem and parametrization

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Parametrization in Green's theorem is often necessary for line integrals over a plane, but its use can depend on the specific problem. While sometimes re-parametrization may be required, the choice to parametrize is generally flexible and aimed at simplifying the original problem. Typically, line integrals are parameterized, while double integrals for bounded regions are not, unless dealing with non-flat surfaces in higher dimensions. Understanding when to apply parametrization can help clarify the process and improve problem-solving efficiency. Ultimately, the decision to parametrize should enhance the ease of calculation in applying Green's theorem.
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Hello. I just wonder if anybody know if there are any rules, when to use parametrization to greens theorem in a vector line integral over a plane. Becouse, it seems sometimes, you have to parametrizice, and other places you dont. I get confused.
 
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It's completely up to you. Of course sometimes you might be asked to re-parametrize or not depending on the problem, but in general it doesn't make a difference. The whole point of parameterizing is to re-write functions in a different way, with the goal of making your parametrization simpler than the problem you began with. So even though one way might be easier than the other, it's totally up to you when to parametrize.
 
Usually you always parameterize the line integral in greens, and don't parameterize the double integral for the region bounded by the line, parameterization for those only comes in handy for not non flat surfaces ie. for the more general Stokes theorem for R3 and up.
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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