Question

1) Show that ?F(x, y) = yî + x? is a conservative vector field using the Curl of ?F(x, y). 2) Show that Line Integral is path independent for the two passes, (1) C1: y = x, (2) C2: y = x^2 from (0,0) to (1,1) in ?F(x, y) = yî + x? C1: y = x gives x = t and y = t for 0 ? t ? 1 C2: y = x^2 gives x = t and y = t^2 for 0 ? t ? 1 TIP: First of all, you should parameterize each path, C1 and C2 to get ?r1(t) and ?r2(t).

          1) Show that ?F(x, y) = yî + x? is a conservative vector field using the Curl of ?F(x, y).
2) Show that Line Integral is path independent for the two passes, (1) C1: y = x, (2) C2: y = x^2 from (0,0) to (1,1) in ?F(x, y) = yî + x?

C1: y = x gives x = t and y = t for 0 ? t ? 1
C2: y = x^2 gives x = t and y = t^2 for 0 ? t ? 1

TIP: First of all, you should parameterize each path, C1 and C2 to get ?r1(t) and ?r2(t).

1) Show that ?F(x, y) = yî + x? is a conservative vector field using the Curl of ?F(x, y).
2) Show that Line Integral is path independent for the two passes, (1) C1: y = x, (2) C2: y = x^2 from (0,0) to (1,1) in ?F(x, y) = yî + x?

C1: y = x gives x = t and y = t for 0 ? t ? 1
C2: y = x^2 gives x = t and y = t^2 for 0 ? t ? 1

TIP: First of all, you should parameterize each path, C1 and C2 to get ?r1(t) and ?r2(t).

Added by Ian H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Vincenzo Zaccaro

12/04/2022

Step 1

The curl of a vector field \( F(x,y) = P(x,y)i + Q(x,y)j \) is given by: \[ \text{curl}(F) = \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)k \] In this case, \( P(x,y) = y \) and \( Q(x,y) = x \). So, we have: \[ \frac{\partial Show more…

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Show that F(x,y) = yi + xj is a conservative vector field using the Curl of F(x,y). Show that the Line Integral is path independent for the two paths, (1) C1: y = x, (2) C2: y = x^2 from (0,0) to (1,1) in F(x,y) = yi + xj. C1: y = x gives x = t and y = t for 0 ≤ t ≤ 1 C2: y = x^2 gives x = t and y = t^2 for 0 ≤ t ≤ 1 TIP: First of all, you should parameterize each path, C1 and C2 to get r1(t) and r2(t).