Question

calc 3

7) Fundamental Theorem of Line Integrals. a) Show that the vector field, F(x,y) = (2x - 2)i - 23e2v j, is conservative. b) Find a potential function for F. c) Evaluate F. dr if C is the path connecting the three line segments from (1,0) to (2,5) then from (2,5) to (-2,5) and finally from (-2,5) to (-1,0).

Answer 1

Q7. Given F = (2x - e 24, î - 2x029 } for conservative vector field - 2824 +2e24 - 0 vector field is conservative. - le given potential af – for function x 824 F. of = – zxe ad e 24 dx - Integrating fase ² - xe + fcy) differentiating aller to y of - 0-2x et fly) - 2x02y = – 20 e 24 + t'cy) fly) = 0 I fly) = 0 If = x²-xe

© le Fi den = s dcxc² x 624) (-25) (215) dx²-xe 2) + C dcx-ne It -Clo) (-1,0) dcxt²x e 24 ) JC-215) €2, 5t (10) = (de 2 - x x 2) (215) + ( xe 2 xx zy (-25) Cat ²- x 24 (-1,0) Jo (-215)

IF.dr = {4-2000 - 1 te od + (4 + 2 0 1 0 - 4 + 2010 ] + lite - 4 - 2010 ] = 4-20 to - X+X+X+ yeto x + 1 + x-zelo - 2
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