Questions: Applications of Multivariable Calculus

Question 1

A particle moves from point A to point B through a gravitational field along two different paths — one short and straight, one long and winding. How do the amounts of work done by gravity along each path compare?

• A. The straight path requires less work because it is shorter
• B. The winding path requires more work because the particle travels farther
• C. Both paths require the same work because gravity is a conservative field
• D. The comparison depends on the particle's mass and speed

Answer: (C)

Gravity is a conservative field — it equals the gradient of a scalar potential (gravitational potential energy). The fundamental theorem for line integrals says that for conservative fields, work depends only on the endpoints: ∫_C ∇φ · dr = φ(end) − φ(start). Both paths from A to B yield the same work. The common mistake is reasoning from physical path length, which is relevant for friction (a non-conservative force) but irrelevant for conservative fields.

Question 2

A closed surface S encloses a volume V. A vector field F has divergence equal to zero at every point inside V. What is the total outward flux of F through S?

• A. It depends on the shape of S and the values of F on the surface
• B. Zero, because the divergence theorem equates flux to the integral of divergence over V
• C. Positive, because flux is always an outward quantity
• D. It cannot be determined without knowing F explicitly on S

Answer: (B)

The divergence theorem states: ∯_S F · dS = ∫∫∫_V (∇·F) dV. If ∇·F = 0 everywhere inside V, the volume integral is zero, so the net flux is zero — regardless of the surface shape or the values of F on the surface. This is the power of the theorem: local information (zero divergence everywhere) translates directly into global information (zero net flux through any enclosing surface).

Question 3

For a conservative vector field in three dimensions, the work done along any closed path is zero.

• T. True
• F. False

Answer: True

This follows directly from the fundamental theorem for line integrals: ∫_C ∇φ · dr = φ(end) − φ(start). On a closed path, start = end, so work = φ(start) − φ(start) = 0. This is why conservative forces can be associated with potential energy — energy lost going 'uphill' is exactly recovered going 'downhill.' Non-conservative forces like friction do not satisfy this; work along a closed path is nonzero.

Question 4

Lagrange multipliers find the global maximum of a function f(x, y) over most of ℝ².

• T. True
• F. False

Answer: False

Lagrange multipliers solve a different problem: finding extrema of f *subject to a constraint* g(x, y) = c, where the search is confined to the constraint curve rather than all of ℝ². Without the constraint, ordinary calculus (setting ∇f = 0) handles unconstrained extrema. The Lagrange method applies when you are forced to stay on a level curve or surface — a budget constraint, fixed distance from a point, a sphere's surface, etc. The geometric insight is that at a constrained extremum, ∇f must be parallel to ∇g.

Question 5

Explain why the condition curl F = 0 is related to path-independence of work, and what this has to do with the existence of a potential function.

Model answer: If F = ∇φ (a conservative field), then curl(∇φ) = 0 always — the curl of any gradient is identically zero. By Stokes' theorem, ∮_C F · dr = ∫∫_S (curl F) · dS; if curl F = 0, the circulation around any closed curve is zero, so work is path-independent. Conversely (in simply connected domains), curl F = 0 guarantees F = ∇φ for some φ. The potential φ is the accounting device that makes path-independence possible: work equals the potential difference at the endpoints, regardless of path.

The connection curl F = 0 ↔ path-independence ↔ existence of potential is the central theorem of vector calculus applications. Gravity and electrostatics are conservative (curl E = 0 in statics), which is why gravitational and electrical potential energy are well-defined quantities. Magnetic fields from steady currents have nonzero curl (Ampère's law: curl B = μ₀J), explaining why magnetic work cannot be stored as simple potential energy in the same way.