Fundamental Theorem for Line Integrals Problems
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Fundamental Theorem for Line Integrals Problems

[From: ] [author: ] [Date: 12-06-29] [Hit: ]
d (vector) s where B is the path from P to Q along the bottom half of that ellipse.-Note that ∫c grad φ · dr = 0 for any closed loop c.Letting C be the ellipse 3x^2+11y^2 = 23 with counterclockwise orientation,==> 0 = ∫B grad φ · dr - ∫T grad φ · dr,==> ∫B grad φ · dr = 1.I hope this helps!......
Suppose ∫ (T on bottom) ∇φ .d (vector) s= 1 where T is the path from P = (-2; 1) to Q = (2; 1)
along the top half of the ellipse 3x^2+11y^2 = 23. Determine the value of ∫ (B on bottom) ∇φ .d (vector) s where B is the path from P to Q along the bottom half of that ellipse.

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Note that ∫c grad φ · dr = 0 for any closed loop c.

Letting C be the ellipse 3x^2+11y^2 = 23 with counterclockwise orientation, we have
0 = ∫c grad φ · dr = ∫c(lower) grad φ · dr + ∫c(higher) grad φ · dr
==> 0 = ∫B grad φ · dr - ∫T grad φ · dr, due to orientation
==> 0 = ∫B grad φ · dr - 1
==> ∫B grad φ · dr = 1.

I hope this helps!
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keywords: Integrals,Theorem,for,Line,Fundamental,Problems,Fundamental Theorem for Line Integrals Problems
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