6.1 Line Integrals of Vector Fields

Line integrals of vector fields measure work done by forces along paths. They're calculated using parametrization and dot products. Understanding these integrals is key to grasping how forces interact with objects moving through space.

The Fundamental Theorem for line integrals links conservative fields to potential functions. This connection simplifies calculations and reveals important properties of certain force fields, like path independence. It's a powerful tool for analyzing physical systems.

Line Integrals of Vector Fields

Line integrals of vector fields

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• Line integral of a vector field quantifies work done by force field $\mathbf{F}$ along curve $C$ expressed as $\int_C \mathbf{F} \cdot d\mathbf{r}$
• Parametrization of curves represents path as function of parameter $t$ (time)
  • 2D: $\mathbf{r}(t) = (x(t), y(t))$
  • 3D: $\mathbf{r}(t) = (x(t), y(t), z(t))$
• Computation process involves substituting parametrization, calculating dot product, and integrating
• Two-dimensional case calculated using $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b [P(x(t),y(t))x'(t) + Q(x(t),y(t))y'(t)] dt$
• Three-dimensional case extends formula to include z-component $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b [P(x(t),y(t),z(t))x'(t) + Q(x(t),y(t),z(t))y'(t) + R(x(t),y(t),z(t))z'(t)] dt$

Fundamental theorem for line integrals

• Conservative vector fields exhibit path-independent line integrals (electric field)
• Potential function $f$ relates to conservative field through gradient $\mathbf{F} = \nabla f$
• Fundamental Theorem states $\int_C \mathbf{F} \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))$ for conservative fields
• Application process:

1. Identify conservative field
2. Find potential function
3. Evaluate potential at endpoints
4. Calculate difference

Conservative vector fields vs line integrals

• Path independence test evaluates line integral over closed path (zero result suggests conservativeness)
• Simply connected regions without holes allow closed path test to be sufficient (circular region)
• Multiple path test compares line integrals between two points along different paths (equal results indicate conservativeness)
• Curl test for 3D fields calculates $\nabla \times \mathbf{F}$ (zero curl everywhere confirms conservative field)

Potential functions for conservative fields

• Partial derivative method sets up equations $\frac{\partial f}{\partial x} = P$, $\frac{\partial f}{\partial y} = Q$, $\frac{\partial f}{\partial z} = R$, then integrates and combines results
• Line integral method chooses convenient path from origin to $(x,y,z)$, computes integral along path to find potential function
• Fundamental Theorem approach uses $f(\mathbf{r}) - f(\mathbf{r}_0) = \int_C \mathbf{F} \cdot d\mathbf{r}$, integrating from reference point $\mathbf{r}_0$ to general point $\mathbf{r}$