- #1

- 375

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- Homework Statement
- https://mathhelpforum.com/attachments/1667128375169-png.45062/

- Relevant Equations
- ##F=\langle z,x,y \rangle, z = 2x + 2y -1## and ##z = x^2 + y^2 ##

My answer:

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- Thread starter WMDhamnekar
- Start date

In summary, the conversation is about calculating integrals using parametrization and verifying the use of Stokes' theorem in a specific case. The expert suggests forgetting about the theorem and just calculating the integrals, and also provides a suggestion for correcting an incorrect parametrization. The conversation also includes a note on using correct formatting in LaTeX.f

- #1

- 375

- 28

- Homework Statement
- https://mathhelpforum.com/attachments/1667128375169-png.45062/

- Relevant Equations
- ##F=\langle z,x,y \rangle, z = 2x + 2y -1## and ##z = x^2 + y^2 ##

My answer:

- #2

- 20,004

- 10,663

To answer your question: You would just compute the integrals as usual.

- #3

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So forget about using the theorem and just calculate the other integrals.

- #4

- 9

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Just curious, which book is this problem from?

- #5

- 375

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I computed line integral as follows:

So forget about using the theorem and just calculate the other integrals.

Curve ##D = (x-1)^2 + (y-1)^2 = 1## Now I parametrize it ##x = \cos(t) , y= \sin(t) , z = 2\cos(t) + 2\sin(t) ##

##F (x,y,z) = zdx + xdy +y dz, r(t) = (\cos(t))i + (\sin(t))j +(\cos(t) +\sin(t))k , x'(t) = - \sin(t), y'(t)= \cos(t), z'(t)= -2\sin(t) +2\cos(t) ##

##\begin{align*}\displaystyle\int_ D F\cdot dr &= \displaystyle\int_0^{2\pi} (2\cos(t)+ 2\sin(t))(-\sin(t)) +(\cos(t))(\cos(t)+(\sin(t))(-2\sin(t) + 2\cos(t))dt\\

&= \displaystyle\int_0^{2\pi} \cos^2(t)-4\sin^2(t)-4\sin(t)\cos(t) dt\\

&= -3\pi \end{align*}##

Last edited:

- #6

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Your parametrization does not describe the correct curve.

- #7

- 375

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If my parametrization of the curve is wrong, what is your suggestion for correct parametrization of the curve?Your parametrization does not describe the correct curve.

- #8

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I believe it would be more instructive if you first verified that it is indeed wrong and then think about what you could do to correct it. What do you get if you insert the parametrization in, for example, the equation for the paraboloid ##z = x^2 + y^2## or the plane ##z = 2x + 2y - 1##?If my parametrization of the curve is wrong, what is your suggestion for correct parametrization of the curve?

Edit: Or indeed into ##(x-1)^2 + (y-1)^2 = 1## ...

Edit 2: On a TeXnical note: Please use \cos and \sin for the trigonometric functions in LaTeX. Just writing sin and cos makes LaTeX typest it as if it was a product of ##s##, ##i##, and ##n## and ##c##, ##o##, and ##s##, respectively. Note the difference:

$$

sin x \leftrightarrow \sin x \qquad cos x \leftrightarrow \cos x

$$

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