MATH 0240 — Analytic Geometry and Calculus 3

MATH 0240 Study Guide

Analytic Geometry and Calculus 3

Multivariable calculus extends rates and accumulation to several variables and to curves, surfaces, and volumes. Pitt's published schedule includes vectors, partial derivatives, multiple integrals, coordinate changes, vector fields, line and surface integrals, and Green's, Stokes', and divergence theorems. Scope: Pitt course guidelines and published topic schedule.

These original notes and examples prioritize geometric meaning, setup, and theorem conditions. Formulas are editable text. In this guide, boldface is not required to identify vectors: r, F, a, and b are vectors when described that way; · means dot product and × means cross product.

Core topic map

Geometry in space. Describe lines, planes, surfaces, distances, and angles using vectors and equations.
Vector-valued functions. Parameterize a curve and interpret velocity, acceleration, tangent direction, and arc length.
Partial derivatives. Measure change with some inputs held fixed, then combine changes using the chain rule and gradient.
Optimization. Find interior critical points, classify them when possible, and examine boundaries and constraints.
Multiple integration. Sketch the region, select an integration order, and choose coordinates that simplify geometry.
Vector calculus. Distinguish work along a curve, circulation around a boundary, and flux through a surface.
Integral theorems. Replace difficult integrals with equivalent ones when the geometry, orientation, and hypotheses allow it.

Important vocabulary

Dot product: Scalar measuring directional alignment and projection.
Cross product: Three-dimensional vector perpendicular to two input vectors.
Normal vector: Vector perpendicular to a plane or surface tangent directions.
Parameterization: Representation of a curve or surface by one or more parameters.
Level curve: Set of points where a two-variable scalar function has a constant value.
Level surface: Three-dimensional set where a scalar function has a constant value.
Partial derivative: Derivative with respect to one variable while holding others fixed.
Gradient: Vector of first partial derivatives of a scalar function.
Directional derivative: Rate of change along a specified unit direction.
Tangent plane: Local linear approximation to a differentiable surface.
Critical point: Point where the gradient vanishes or where relevant derivatives fail to exist.
Saddle point: Point with increasing behavior in some directions and decreasing behavior in others, rather than a local maximum or minimum.
Lagrange multiplier: Auxiliary scalar used in constrained optimization conditions.
Iterated integral: Multiple integral evaluated through successive one-variable integrals.
Jacobian: Determinant describing local area or volume scaling under a coordinate transformation.
Vector field: Assignment of a vector to each point of a region.
Line integral: Accumulation along a curve.
Conservative field: Vector field equal to the gradient of a scalar potential on its domain.
Circulation: Tangential vector-field accumulation around a closed curve.
Flux: Oriented flow of a vector field through a surface.
Divergence: Scalar measuring local outward-flow density.
Curl: Vector measuring local rotational tendency.
Orientation: Consistent choice of traversal direction or surface normal.
Simply connected domain: Roughly, a connected domain where loops can be continuously contracted to a point within the domain.

Formula reference

Vectors and space curves

|a| = √(a_{1}^{2} + a_{2}^{2} + a_{3}^{2}); unit vector in a's direction is a/|a| for a ≠ 0.
a · b = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} = |a||b|cosθ.
a × b = ⟨a_{2}b_{3} − a_{3}b_{2}, a_{3}b_{1} − a_{1}b_{3}, a_{1}b_{2} − a_{2}b_{1}⟩.
|a × b| = |a||b|sinθ gives parallelogram area.
Line: r(t) = r_{0} + tv, with nonzero direction v.
Plane: n · (r − r_{0}) = 0, with nonzero normal n.
Velocity v = r′(t); acceleration a = r″(t); speed = |r′(t)|.
Arc length L = ∫_{a}^{b}|r′(t)| dt.
Curvature κ = |r′ × r″|/|r′|^{3} for a sufficiently smooth regular space curve.

Differentiation and optimization

Gradient ∇f = ⟨f_{x}, fᵧ, f_z⟩ in three dimensions.
Directional derivative Dᵤf = ∇f · u for a unit vector u and differentiable f.
Tangent plane to z = f(x,y) at (a,b): z = f(a,b) + f_{x}(a,b)(x − a) + fᵧ(a,b)(y − b).
Along x(t), y(t): df/dt = f_{x} dx/dt + fᵧ dy/dt.
Linear change estimate: Δf ≈ f_{x}Δx + fᵧΔy, evaluated at the base point.
Two-variable second-derivative test: D = f_{xx}fᵧᵧ − (f_{x}ᵧ)^{2} at a critical point, assuming continuous second derivatives nearby. D > 0 with f_{xx} > 0 gives a local minimum; D > 0 with f_{xx} < 0 gives a local maximum; D < 0 gives a saddle; D = 0 is inconclusive.
One smooth constraint g = c: ∇f = λ∇g at a regular constrained extremum. Also satisfy g = c and inspect boundary or singular cases separately.

Multiple integrals and coordinates

Area of D = ∬D 1 dA; volume of E = ∭E 1 dV.
Mass with density ρ: m = ∬Dρ dA for a lamina or ∭Eρ dV for a solid.
Center-of-mass coordinate x̄ = (1/m)∭E xρ dV, with analogous formulas for y and z and matching two-dimensional versions.
Polar: x = r cosθ, y = r sinθ; dA = r dr dθ.
Cylindrical: x = r cosθ, y = r sinθ, z = z; dV = r dr dθ dz, reordered consistently with bounds.
Spherical convention here: x = ρ sinφ cosθ, y = ρ sinφ sinθ, z = ρ cosφ; φ is measured from +z and θ around the xy plane. dV = ρ^{2}sinφ dρ dφ dθ.
Two-dimensional change of variables: dA = |∂(x,y)/∂(u,v)| du dv. Use the absolute determinant and the transformed region.

Line integrals, flux, and theorems

Scalar line integral ∫C f ds = ∫_{a}^{b}f(r(t))|r′(t)| dt.
Work integral ∫C F · dr = ∫_{a}^{b}F(r(t)) · r′(t) dt.
If F = ∇φ, then ∫C F · dr = φ(endpoint) − φ(startpoint).
For F = ⟨P,Q,R⟩: div F = P_{x} + Qᵧ + R_z.
curl F = ∇×F = ⟨Rᵧ − Q_z, P_z − R_{x}, Q_{x} − Pᵧ⟩.
Green's theorem: ∮C(P dx + Q dy) = ∬D(Q_{x} − Pᵧ) dA for a positively oriented simple closed boundary and continuously differentiable field on a region containing D. Hole boundaries require consistent orientation.
Surface area for r(u,v): ∬|rᵤ × rᵥ| du dv.
Flux through a parameterized surface: ∬S F · n dS = ∬F(r(u,v)) · (rᵤ × rᵥ) du dv, with the cross product chosen to give the required orientation.
Stokes' theorem: ∮∂S F · dr = ∬S(curl F) · n dS, with compatible right-hand-rule boundary orientation and the required smoothness.
Divergence theorem: ∬∂E F · n dS = ∭E div F dV for a closed boundary with outward normal and a sufficiently smooth field throughout the solid.
A continuously differentiable conservative field has zero curl. The converse requires domain conditions, such as a suitable simply connected open domain; zero curl alone is insufficient on arbitrary domains.

Worked examples

Example 1 Directional derivative

Let f(x,y) = x^{2}y at (1,2), moving in the direction ⟨3,4⟩. The unit direction is u = ⟨3/5,4/5⟩. The gradient is ⟨2xy,x^{2}⟩, giving ⟨4,1⟩ at the point. Dᵤf = 4(3/5) + 1(4/5) = 16/5. Omitting normalization would give a rate scaled by the direction vector's length.

Example 2 Polar integration

Integrate x^{2} + y^{2} over the unit disk. Use x^{2} + y^{2} = r^{2}, 0 ≤ r ≤ 1, and 0 ≤ θ ≤ 2π. The integral becomes ∫{0}^{2}π∫{0}^{1}r^{2}(r dr dθ) = 2π(1/4) = π/2. The extra r is the area-scaling factor, not an optional part of the integrand.

Example 3 Conservative field

For F = ⟨2xy,x^{2}⟩, a potential is φ = x^{2}y because ∇φ reproduces both components. Work from (0,0) to (1,2) along any suitable curve in the plane is φ(1,2) − φ(0,0) = 2. A potential is stronger evidence than checking one convenient path.

Example 4 Closed-surface flux

Let F = ⟨x,y,z⟩ on the sphere of radius a centered at the origin. div F = 3. The outward flux is ∭E3 dV = 3(4πa^{3}/3) = 4πa^{3}. The theorem applies because the surface is closed and the field is smooth throughout the ball.

Practice questions and answers

∇f for f = x^{2} + y^{2} + z^{2}? ⟨2x,2y,2z⟩.
A plane through (1,0,0) with normal ⟨2,3,4⟩? 2(x − 1) + 3y + 4z = 0.
div⟨x^{2},y,z⟩? 2x + 2.
What is ∮C∇φ · dr around a closed curve where φ is defined and smooth? 0.
Find critical points of f = x^{2} + y^{2} − 2x − 4y. (1,2), a strict minimum.
What changes if a work-integral path orientation reverses? The sign reverses.
Does reversing orientation change a scalar line integral with ds? No; ds is unsigned arc length.

Mistakes to catch

Forgetting to normalize a directional vector.
Omitting Jacobians in polar, cylindrical, or spherical integration.
Writing bounds without sketching the region or checking which variable depends on another.
Using the divergence theorem on an open surface without accounting for a closing surface.
Forgetting boundary orientation in Green's or Stokes' theorem.
Treating a zero Hessian determinant as proof that no extremum exists.
Applying curl tests across a singularity or hole without checking the domain.

Review routine

For every integral, identify the object being integrated over, the quantity being accumulated, the orientation if any, and the differential element. Draw the region before choosing coordinates. Practice calculating one problem both directly and with an integral theorem so you can see why the answers agree.