Vectors Prep
For Vector Analysis MATH 4551
From Advanced Calculus - Kaplan, 5th Edition
Foundations
Prerequisite refreshers - start here if you're rustyTopology & Domains
Partial Derivatives & Chain Rule
Line Integrals Intro
Definite Integrals & the FTC
Improper Integrals
Integration Techniques
Topics
Learning material with interactive lessonsChapter 2: Differential Calculus of Several Variables
Total Differential & the Fundamental Lemma
The right notion of differentiability for several variables - and the lemma that justifies the chain rule.
The Jacobian Matrix & the Differential
The matrix of the differential - the chain rule as matrix multiplication, determinant as local volume scaling.
The General Chain Rule
Jacobian matrices multiply under composition - area distortion factors compose.
Implicit Functions
When equations secretly define functions - the derivative formula and the Implicit Function Theorem.
Proof of the IFT
How monotonicity and the IVT force a unique smooth solution into existence.
Inverse Functions & Coordinates
Coordinate transformations, the Jacobian reciprocal rule, and polar/cylindrical/spherical systems.
Higher Derivatives of Implicit Functions
Find second-order partial derivatives without solving for z explicitly.
Maxima & Minima
Find and classify critical points - the second derivative test, saddle points, and absolute extrema.
Quadratic Forms & Definiteness
Classify critical points using the Hessian, eigenvalues, and quadratic forms.
Chapter 3: Vector Differential Calculus
Introduction
The fluid model, stream lines, and a first taste of divergence and curl.
Vector Fields & Scalar Fields
Radial, rotation, and gravitational fields. Scalar fields and level curves.
The Gradient Field
Directional derivatives, steepest ascent, and the geometry of level curves.
Divergence
Sources, sinks, and net flux - measuring how much a field spreads out.
The Curl
Rotation in a vector field - the paddle wheel test and the curl formula.
Combined Operations
Chaining grad, div, and curl - fundamental identities, the Laplacian, and potential functions.
Finding Potential Functions
The three-integration recipe for recovering $f$ from $\nabla f = \mathbf{u}$, with traps and a worked 3D example.
Chapter 4: Integral Calculus of Several Variables
The Definite Integral
Riemann sums, iterated integrals, and changing the order of integration.
Numerical Evaluation of Indefinite Integrals
Building functions one trapezoid at a time, and elliptic integrals.
Double Integrals
Integrating over curved regions, physical applications, and the mean value property.
Triple Integrals
Extending integration to three dimensions - mass, volume, and iterated integrals.
Integrals of Vector Functions
Component-wise integration, displacement, and work along a path.
Change of Variables
Jacobian determinants, coordinate transformations, and area stretching.
Arc Length & Surface Area
Measuring curves and surfaces via parametric integrals and cross products.
Improper Multiple Integrals
Convergence over unbounded regions and near singularities.
Leibnitz's Rule
Differentiate under the integral sign with fixed or variable limits.
Chapter 5: Line Integrals & Green's Theorem
Introduction to Vector Integral Calculus
Three kinds of line integrals: arc length, mass, and work along a curve.
Line Integrals in the Plane
Parametric computation of ∫ P dx + Q dy, direction, closed curves, piecewise paths.
Line Integrals
Scalar line integrals with respect to arc length, curtain visualization.
Line Integrals as Integrals of Vectors
The vector form $\int_C \mathbf{u}\cdot d\mathbf{r}$ - work, circulation, and the conservative-field shortcut.
Green's Theorem
Convert closed-curve line integrals into double integrals over the enclosed region.
Path Independence & the Four Theorems
When the line integral depends only on endpoints - and the precise hypotheses (open connected vs simply connected) that decide it.
Multiply Connected Domains
Domains with holes: the deformation principle, winding numbers, and how to integrate around singularities.
Surface Integrals
Integrate scalars and vector fields over a curved surface - the prerequisite for Divergence and Stokes.
The Divergence Theorem
Flux across a closed surface equals the total divergence inside - Gauss's theorem.
Stokes's Theorem
Circulation around a boundary curve equals the total curl through any spanning surface.
Path Independence in 3D
Four theorems for line integrals in space - and why removing a line is worse than removing a point.