# How did Griffith check Stoke's theorem in this case?

• garylau
In summary, Griffith was able to get this long-horrible equation by using trigonometric substitution. It isn't tedious at all, if you use the proper trigonometric substitution (you can even do it in your head after some training, though the paper is still less error-prone).
garylau
<Moderator's note: Moved from a technical forum, so homework template missing>

Sorry
i have one question to ask
how to check the v.dl part in this problem
i cannot do this problem as it is too hard to integrate the equation

How did griffith get this long-horrible equation(see the orange circle)?
it sounds unreasonable and too hard to get
and is it possible that there are
other faster ways to check the v.dl part in this problem?

thank you

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You can just take both integrals (one from each term in the numerator) from the table of integrals. Here's a list of integrals of irrational functions on the wiki. Or you can spend some time trying to derive those integrals yourself using the standard methods (but you don't need to do it each and every time - that's what the tables are for), as a useful exercise. Try to change the variable to some convenient trigonometric function to get rid of those square roots for starters.

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Dragon27 said:
You can just take both integrals (one from each term in the numerator) from the table of integrals. Here's a list of integrals of irrational functions on the wiki. Or you can spend some time trying to derive those integrals yourself using the standard methods (but you don't need to do it each and every time - that's what the tables are for), as a useful exercise. Try to change variables to some convenient trigonometric function to get rid of those square roots for starters.

thank

How did he prove this formula?

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garylau said:
thank
How did he prove this formula?
What kind of integration techniques are you familiar with?

garylau
Dragon27 said:
What kind of integration techniques are you familiar with?
the more simple one
but
i havn't derived this formula(the two circles that i emphasize) before and not familiar with this formula
it looks quiet tedious

do you know what technique the Wiki is using to claim these statements

thank

Dragon27 said:
It isn't tedious at all, if you use the proper trigonometric substitution (you can even do it in your head after some training, though the paper is still less error-prone).
Dragon27 said:
It isn't tedious at all, if you use the proper trigonometric substitution (you can even do it in your head after some training, though the paper is still less error-prone).
thank you
however
Khan didnt derive this formula which included x^2?
where is these Two terms coming from

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Well, if you went through the tutorial, you should be able to derive this by yourself (the substitution is the same).
How about you start doing this integral and show us where you get stuck?

garylau

## 1. How did Griffith check Stoke's theorem in this case?

Griffith checked Stoke's theorem in this case by first verifying that the vector field in question was differentiable and had a continuous first derivative. Then, he found a closed surface that enclosed the curve on which the line integral was being calculated. Finally, he used the divergence theorem to convert the surface integral into a triple integral, which he evaluated to prove that the line integral and surface integral were equal.

## 2. What is Stoke's theorem and why is it important?

Stoke's theorem is a fundamental theorem in vector calculus that relates a line integral around a closed curve to a surface integral over the region enclosed by that curve. It is important because it allows us to easily calculate line integrals by converting them into surface integrals, which are often easier to evaluate.

## 3. What is the difference between Stoke's theorem and the divergence theorem?

Stoke's theorem relates a line integral to a surface integral, while the divergence theorem relates a volume integral to a surface integral. In other words, Stoke's theorem deals with curves and surfaces, while the divergence theorem deals with surfaces and volumes.

## 4. Can Stoke's theorem be used for any vector field?

No, Stoke's theorem can only be used for vector fields that are differentiable and have a continuous first derivative. This ensures that the surface integral can be evaluated using the divergence theorem.

## 5. Are there any limitations to Stoke's theorem?

One limitation of Stoke's theorem is that it only applies to closed curves and surfaces. It cannot be used for open curves or surfaces. Additionally, the vector field must be conservative for the theorem to hold true.

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