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Calculus III (Multivariable) Unit 7: Vector calculus

Vector fields assign a vector to every point. Line integrals add up a field along a curve, which in physics is work.

Covers: Vector fields, line integrals, conservative fields, Green's theorem

Study guide

A vector field F assigns a vector to every point, like wind velocity or a force. In 2D, F = ⟨P, Q⟩; in 3D, F = ⟨P, Q, R⟩. Gradient fields F = ∇f come from a scalar potential function f.

A line integral adds a quantity along a curve C. For a scalar function, ∫_C f ds uses arc length, ds = |r′(t)| dt. For a vector field, ∫_C F·dr = ∫ F(r(t))·r′(t) dt measures work done by a force along a path, or flow along a curve. Parametrize the curve, substitute, and integrate in t. Reversing the direction of C changes the sign.

A conservative field is a gradient field. Its line integrals are path-independent and given by the Fundamental Theorem for line integrals: ∫_C ∇f·dr = f(end) − f(start). Around any closed loop the integral is 0. In 2D, on a simply connected region, F = ⟨P, Q⟩ is conservative when ∂P/∂y = ∂Q/∂x. To find the potential, integrate P with respect to x, then match ∂f/∂y with Q to fix the remaining function of y.

Green's theorem converts a line integral around a closed curve into a double integral over the region inside: ∮_C P dx + Q dy = ∬_D (∂Q/∂x − ∂P/∂y) dA, with C oriented counterclockwise. It can also compute areas, for example using ½∮ x dy − y dx.

The outward flux of F = ⟨P, Q⟩ across a closed curve C is ∮_C F·n ds = ∮_C P dy − Q dx = ∬_D (∂P/∂x + ∂Q/∂y) dA, with C traversed counterclockwise. This is the 2D divergence theorem. For a curve traversed counterclockwise, the outward normal times ds is ⟨dy, −dx⟩. Along any segment where F is parallel to the curve, the flux is 0.

In 3D, F = ⟨P, Q, R⟩ is conservative on a simply connected region when curl F = 0, which means R_y = Q_z, P_z = R_x and Q_x = P_y. A positive net outward flux means more of the field leaves the region than enters it (a net source). A positive circulation means a net counterclockwise flow around the curve.

Key ideas

• Line integral of a scalar function: ∫_C f ds.
• Work: ∫_C F·dr.
• A conservative field is a gradient F = ∇f; its line integrals depend only on the endpoints.
• Test in 2D: F = ⟨P, Q⟩ is conservative on a simply connected region if ∂P/∂y = ∂Q/∂x.
• Green's theorem turns a line integral around a closed curve into a double integral over its interior.

Formulas and rules

Common mistakes

• Using Green's theorem on a curve that isn't closed.
• Going clockwise when Green's theorem assumes counterclockwise.

Words to know

Work done by F along C

∫_C F·dr

Conservative vector field

The gradient of a potential function f; path-independent.

2D test for conservative F = ⟨P, Q⟩

∂P/∂y = ∂Q/∂x (on a simply connected region).

Green's theorem

∮_C P dx + Q dy = ∬_D (Q_x − P_y) dA

∮ of a conservative field around a closed loop

0

Fundamental theorem for line integrals

∫_C ∇f·dr = f(end) − f(start)

Vector field

A vector assigned to every point, like F = ⟨P, Q⟩.

Gradient field

F = ∇f for a potential function f.

Line integral of a scalar function

∫_C f ds, with ds = |r′(t)| dt.

Line integral of a vector field

∫_C F·dr = ∫ F(r(t))·r′(t) dt: work or circulation.

Path independence

A conservative field's line integral depends only on the endpoints.

Potential function

f with ∇f = F.

Area by Green's theorem

½∮ x dy − y dx

Sample practice questions

17 more questions like these, from level 1 to 5, are in the practice set.