Chapter 16 (Vector Fields, Line Integrals, Flux, Green’s, Stokes’, Divergence)

Vector Field

A function that assigns a vector to each point in space.

2D Vector Field

F(x, y) = ⟨P(x, y), Q(x, y)⟩

3D Vector Field

F(x, y, z) = ⟨P, Q, R⟩

Physical Meaning of a Vector Field

Models force, velocity, or flow.

Scalar Line Integral

Integrates a scalar function along a curve.

Scalar Line Integral Formula

∫ f(r(t)) ‖r′(t)‖ dt

What Scalar Line Integrals Measure

Accumulated quantity along a path.

Vector Line Integral

Integrates a vector field along a curve.

Vector Line Integral Formula

∫ F(r(t)) · r′(t) dt

Physical Meaning (Work)

Work done by a force field along a path.

Orientation of a Curve

Direction the curve is traversed.

Effect of Reversing Orientation

Changes the sign of the line integral.

Conservative Vector Field

A vector field that is the gradient of a scalar function.

Potential Function

A scalar function f such that ∇f = F.

Path Independence

Line integral depends only on endpoints.

Test for Conservative Field (2D)

∂P/∂y = ∂Q/∂x (on simply connected domains)

Fundamental Theorem of Line Integrals

∫C F · dr = f(B) − f(A)

Green’s Theorem (Circulation Form)

Converts a line integral into a double integral.

Circulation Form

∮C (P dx + Q dy) = ∬ (∂Q/∂x − ∂P/∂y) dA

Green’s Theorem (Flux Form)

∮C F · n ds = ∬ (∂P/∂x + ∂Q/∂y) dA

When Can You Use Green’s Theorem?

2D, closed curve, positively oriented.

Positive Orientation

Counterclockwise.

Divergence

Measures outward flow from a point.

Divergence Formula (2D)

∇ · F = ∂P/∂x + ∂Q/∂y

Curl (2D)

Measures rotation or circulation.

Curl Formula (2D)

∂Q/∂x − ∂P/∂y

Physical Meaning of Divergence

Source or sink strength.

Physical Meaning of Curl

Tendency to rotate.

Surface Integral

Integrates a scalar function over a surface.

Surface Parameterization

r(u, v) describes a surface.

Surface Integral Formula

∬ f(r(u,v)) ‖rᵤ × rᵥ‖ dudv

Flux

Amount of field passing through a surface.

Flux Integral Formula

∬ F · n dS

Orientation of a Surface

Direction of the normal vector.

Outward Orientation

Normal points away from enclosed volume.

Stokes’ Theorem

Relates line integrals to surface integrals.

Stokes’ Formula

∮C F · dr = ∬ (∇ × F) · n dS

When to Use Stokes’ Theorem

Curl + surface + boundary curve.

Orientation Rule (Right-Hand Rule)

Fingers follow curve, thumb gives normal.

Divergence Theorem

Converts flux over surface to triple integral.

Divergence Theorem Formula

∬S F · n dS = ∭V ∇ · F dV

When to Use Divergence Theorem

Closed surface enclosing a volume.

Use Line Integrals When

Integrating along a curve.

Use Green’s Theorem When

2D closed curve + region.

Use Stokes’ Theorem When

Surface with a boundary curve.

Use Divergence Theorem When

Closed surface + volume.

Forgetting Orientation

Can flip the sign of integrals.

Using Wrong Theorem Dimension

Green’s is 2D, Stokes’ is surface, Divergence is volume.

Forgetting Simply Connected Requirement

Conservative tests may fail.