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jeff1evesque
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Homework Statement
Given the attached picture, Calculate the work required to go around the contour shown if the force is [tex]\vec{F} = y\hat{x} - x^{2} \hat{y}[/tex]First by Contour integration:
Work = [tex]\oint \vec{F} \bullet \vec{dl} = \int_{a}^{b} (\vec{F} \bullet \hat{x})dx + \int_{b}^{c} (\vec{F} \bullet \hat{y})dy - \int_{c}^{d} (\vec{F} \bullet \hat{x})dx - \int_{d}^{a} (\vec{F} \bullet \hat{y})dy [/tex]
[tex]= \int_{1}^{2} (2) dx + \int_{2}^{3} (4)dy - \int_{2}^{1} (3)dx - \int_{3}^{2} (1)dy = 0 [/tex]
I think the answer above is suppose to be -4 not 0.Now by Stokes Theorem:
When this was done with "Stokes theorem", we get the following:
Work = [tex]\oint \vec{F} \bullet \vec{dl} \equiv \int \int (\nabla \times \hat{F})\bullet \vec{ds} = - \int \int (2x + 1)( \hat{z} \bullet \hat{z})ds = \int_{2}^{3} \int_{1}^{2} (2x + 1)dxdy = -4[/tex]
Note: [tex]( \hat{z} \bullet \hat{z}) = 1[/tex] since [tex]( \hat{z} \bullet \vec{ds}) = ds [/tex]
Question:
When I look at this, Stokes method seems apparently correct, but the method before it seems wrong. Can someone help me find my error?Thanks,JL
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