5.1 Vector Fields and Line Integrals

Vector fields assign vectors to points in space, representing physical quantities like force or velocity. They're visualized using arrow plots in 2D or 3D, with components defined by scalar functions in each direction.

Line integrals measure accumulated effects along curves in vector fields. They're computed by parameterizing curves and evaluating single-variable integrals. The Fundamental Theorem of Line Integrals simplifies calculations for conservative fields, while work in force fields is calculated using line integrals.

Vector Fields

Vector fields in multiple dimensions

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• Vector field assigns vector to each point in space representing physical quantities (force, velocity, electric field)
• Two-dimensional vector fields expressed as $\mathbf{F}(x, y) = P(x,y)\mathbf{i} + Q(x,y)\mathbf{j}$ visualized using arrow plots on 2D plane
• Three-dimensional vector fields expressed as $\mathbf{F}(x, y, z) = P(x,y,z)\mathbf{i} + Q(x,y,z)\mathbf{j} + R(x,y,z)\mathbf{k}$ visualized using arrow plots in 3D space
• Components of vector fields defined by scalar functions in each direction
• Continuity and smoothness of vector fields ensure no abrupt changes or discontinuities
• Vector fields applied in physics and engineering (electromagnetic fields, fluid flow, gravitational fields)

Line Integrals

Line integrals along curves

• Line integrals integrate along curve or path in vector field measuring accumulated effect
• Curves parameterized as function of parameter $t$ to describe path mathematically
• Line integral formulas: scalar fields $\int_C f(x,y,z) ds$, vector fields $\int_C \mathbf{F} \cdot d\mathbf{r}$
• Computation process:
  1. Parameterize curve
  2. Express integrand in terms of parameter
  3. Evaluate resulting single-variable integral
• Line integrals independent of parameterization choice
• Orientation of curves affects sign of line integral result

Fundamental theorem of line integrals

• Theorem states $\int_C \nabla f \cdot d\mathbf{r} = f(b) - f(a)$ for conservative vector fields
• Conservative vector fields have path-independent line integrals
• Potential functions found for conservative fields by integrating components
• Theorem applications simplify calculations and prove path independence
• Curl test in two and three dimensions determines if field is conservative

Work in force fields

• Work calculated as line integral $W = \int_C \mathbf{F} \cdot d\mathbf{r}$ in force fields
• Force fields in physics include gravitational, electromagnetic, and spring forces
• Calculation process involves identifying force field and curve, setting up line integral, evaluating using parameterization or Fundamental Theorem
• Positive work increases energy of system, negative work decreases energy
• Work path-dependent in non-conservative fields (friction)
• Energy conserved in conservative force fields (gravitational potential energy)