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.
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.
-
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!
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!