ECE 206: Advanced Calculus 2 for Electrical Engineers

Sources and References

Equivalent UW courses — MATH 217 (Calculus 3 for Chemical Engineering), MATH 237/247 (Calculus 3 / Advanced Calculus 1 for Honours Math), AMATH 231 (Calculus 4 / Vector Calculus) Primary textbook — Dennis G. Zill, Advanced Engineering Mathematics, 7th ed., Jones & Bartlett, 2020. Supplementary references — Edward B. Saff and Arthur David Snider, Fundamentals of Complex Analysis with Applications to Engineering and Science, 3rd ed., Pearson, 2003; Jerrold E. Marsden and Anthony J. Tromba, Vector Calculus, 6th ed., W. H. Freeman, 2012.

Equivalent UW Courses

ECE 206 straddles two separate Math offerings. Its vector-calculus half — triple integrals in cylindrical / spherical coordinates, line and surface integrals, and the Green / Gauss / Stokes theorems — is the core of AMATH 231 and the back half of MATH 237/247. The complex-analysis half — analytic functions, Cauchy-Riemann, contour integrals, Laurent series and residues — is normally its own course (AMATH 332 or PMATH 352) in the Math Faculty, and is not covered in MATH 237 at all. MATH 217 packs similar vector calculus for Chemical Engineers but again omits complex analysis. So a Math student would need roughly MATH 237 plus AMATH 231 plus AMATH 332 to match the scope.

What This Course Adds Beyond the Equivalents

• Physical / EM framing. Divergence, curl, and the integral theorems are introduced with Maxwell’s equations as the motivating application, rather than as pure differential geometry. Conformal mapping is tied directly to electrostatic potential problems.
• Complex analysis in one term with vector calculus. The ECE version compresses what Math treats as two separate courses, which means less time on proofs (Jordan curve theorem, uniform convergence of series) and more time on computation and engineering use cases (residues for inverse Laplace transforms, stability, control).
• Omits, relative to MATH 237/247: rigorous multivariable differentiability, the implicit and inverse function theorems, and careful treatment of change-of-variables. Relative to AMATH 332: contour-deformation theorems are stated rather than proved, and there is no discussion of analytic continuation or the Riemann sphere.

Topic Summary

Parametric Curves and Surfaces, Triple Integrals

Quick review of parametrizing curves \( \mathbf r(t) \) and surfaces \( \mathbf r(u,v) \). Iterated integrals in polar, cylindrical, and spherical coordinates; Jacobian for change of variables is used but not derived in full generality.

Line Integrals and Vector Fields

Scalar and vector line integrals, work integrals \( \int_C \mathbf F \cdot d\mathbf r \), and the fundamental theorem for conservative fields. Tests for path-independence via \( \nabla \times \mathbf F = \mathbf 0 \) on simply-connected domains.

Surface Integrals, Divergence, Curl

Oriented surfaces and flux \( \iint_S \mathbf F \cdot d\mathbf S \). Geometric and coordinate formulas for \( \nabla \cdot \mathbf F \) and \( \nabla \times \mathbf F \), with brief intuition (source density, infinitesimal circulation).

Green’s, Divergence, and Stokes’ Theorems

The three integral theorems are presented as one family relating a domain integral to a boundary integral. Direct application to Maxwell’s equations: Gauss’s law, Ampere’s law, Faraday’s law in integral form.

Complex Numbers and Functions

Review of \( \mathbb C \), polar form, roots of unity. Complex functions as mappings; visualising \( w = f(z) \). Elementary functions \( e^z \), \( \log z \), branches.

Analytic Functions and Conformal Mapping

Complex differentiability, the Cauchy-Riemann equations, and the link to harmonic functions. Conformal mappings — Möbius transformations, \( z^2 \), \( \sin z \) — used to solve Laplace’s equation for 2D electrostatic potentials.

Contour Integration

Path integrals in \( \mathbb C \); the Cauchy-Goursat theorem on simply-connected domains; Cauchy’s integral formula and its consequences for derivatives of analytic functions.

Taylor / Laurent Series and Residues

Power series around regular points, Laurent series around isolated singularities, classification of poles vs essential singularities. The residue theorem

\[ \oint_C f(z)\,dz = 2\pi i \sum_k \operatorname{Res}(f, z_k) \]

applied to real improper integrals, trigonometric integrals, and inverse Laplace / Fourier transforms arising in control and signal problems.