Questions: If these paths exist in a conservative vector field F, which vector line integrals will have the same value? Select two or more common integrals. Select all that apply: ∫C2 F · ds+∫C3 F · ds ∫C2 F · ds ∫C1 F · ds ∫C3 F · ds

If these paths exist in a conservative vector field F, which vector line integrals will have the same value? Select two or more common integrals.

Select all that apply:
∫C2 F · ds+∫C3 F · ds
∫C2 F · ds
∫C1 F · ds
∫C3 F · ds

Transcript text: If these paths exist in a conservative vector field $\mathbf{F}$, which vector line integrals will have the same value? Select two or more common integrals. Select all that apply: $\int_{C_{2}} \mathbf{F} \cdot d \mathbf{s}+\int_{C_{3}} \mathbf{F} \cdot d \mathbf{s}$ $\int_{C_{2}} \mathbf{F} \cdot d \mathbf{s}$ $\int_{C_{1}} \mathbf{F} \cdot d \mathbf{s}$ $\int_{C_{3}} \mathbf{F} \cdot d \mathrm{~s}$

Solution

Solution Steps

Step 1: Understanding Conservative Vector Fields

In a conservative vector field, the line integral between two points is independent of the path taken. This means that if we have two paths with the same start and end points, the line integrals along those paths will be equal.

Step 2: Analyzing the Given Paths

We are given paths C₁, C₂, and C₃. Notice that C₂ and C₃ together form a path from the bottom point to the top point. C₁ is another path between those same two points.

Step 3: Comparing the Line Integrals

Since the field is conservative, the integral over C₁ must be equal to the integral over the combined path C₂ and C₃. Mathematically, this means:

∫_C₁ F ⋅ ds = ∫_C₂ F ⋅ ds + ∫_C₃ F ⋅ ds