Mathematical methods

Map of content for the applied math that engineering builds on — the vector calculus and complex analysis that show up everywhere downstream. The path: complex numbers → vectors and coordinate systems → vector calculus (fields, line integrals, surface integrals, integral theorems) → complex analysis (analytic functions, contour integration, residues, conformal mapping).

Complex numbers

The number system that closes the algebra of polynomials and powers every transform method.

• Imaginary unit — $i^2=-1$; the move that closes the reals.
• Algebra and Geometry in the complex plane — addition, multiplication, modulus, argument.
• Complex conjugate — reflect across the real axis.
• Polar representation of complex numbers — $z=r{e}^{j\theta }$; the form multiplication and powers live in.
• Roots of complex numbers — $n$ equally-spaced roots on a circle.
• Fundamental theorem of algebra — every degree-$n$ polynomial has $n$ complex roots.
• Triangle inequality — $|z_1+z_2|\le |z_1|+|z_2|$.
• Complex sequence — convergence in the complex plane.
• Euler’s formula — $e^{j\theta }=\cos \theta +j\sin \theta$; the bridge to trigonometry.
• Complex sinusoid — $A{e}^{j(\omega t+\varphi )}$, the rotating phasor in motion.
• Phasor — frozen-time complex amplitude $A{e}^{j\varphi }$.
• Phasor transform — moving a sinusoidal circuit into the complex plane.
• Phasor relationships for circuit elements — impedances of $R$, $L$, $C$.

Vectors and coordinate systems

Geometric foundations for everything that follows.

• Vector (geometric) — magnitude and direction.
• Unit vector — direction-only.
• Dot product — projection and angle.
• Cross product — perpendicular vector with magnitude $|\mathbf{a}||\mathbf{b}|\sin \theta$.
• Scalar triple product — volume of a parallelepiped.
• Vector triple product — the BAC-CAB identity.
• Cartesian coordinates — the default $(x,y,z)$ frame.
• Cylindrical coordinates — $(r,\varphi ,z)$; natural for axial symmetry.
• Spherical coordinates — $(\rho ,\theta ,\varphi )$; natural for radial symmetry.

Vector-valued functions

Curves in space — the path for line integrals.

• Vector-valued function — $\mathbf{r}(t)$, a parameterized curve.
• Derivative of vector-valued function — componentwise differentiation.
• Unit tangent vector — $\mathbf{T}={\mathbf{r}}^{\prime }/|{\mathbf{r}}^{\prime }|$.
• Tangent line to a curve — local linear approximation.
• Velocity, speed, and acceleration — the kinematic interpretation.
• Arc length — $\int |{\mathbf{r}}^{\prime }(t)|dt$.
• Smooth curve — ${\mathbf{r}}^{\prime }$ continuous and nonzero.

Vector fields and differential operators

Functions that assign a vector (or scalar) to each point in space, and the operators that probe their local structure.

• Vector field — $\mathbf{F}(x,y,z)$ at every point.
• Gradient — direction of steepest ascent of a scalar field.
• Gradient field — vector field that is the gradient of some potential.
• Divergence — net outflow per unit volume.
• Curl — circulation per unit area.
• Stream function — scalar whose level curves are streamlines of a 2D incompressible flow.
• Harmonic function — solution of Laplace’s equation.
• Laplace’s equation — ${\nabla }^2\varphi =0$; the equilibrium PDE.
• Inverse-square field — $\mathbf{F}\propto \hat{\mathbf{r}}/r^2$; the canonical gravitational and electrostatic field.
• Clairaut’s theorem — equality of mixed partials; the exactness test for conservative fields.

Line integrals

Integrating along a curve — work, circulation, flow.

• Line integral — ${\int }_C\mathbf{F}\cdot d\mathbf{r}$.
• Conservative vector field — path-independence, equivalent to having a potential.
• Fundamental theorem of line integrals — ${\int }_C\nabla f\cdot d\mathbf{r}=f(\mathbf{r}(b))-f(\mathbf{r}(a))$.
• Simply connected domain — no holes; lets curl-free imply conservative.

Surface and flux integrals

Integrating over a surface.

• Parameterized surface — $\mathbf{r}(u,v)$; the 2-parameter analog of a curve.
• Scalar surface integral — ${\iint }_{S}fdS$, area-weighted average.
• Flux integral — ${\iint }_S\mathbf{F}\cdot d\mathbf{S}$, the flow through a surface.

Integral theorems

The grand identities linking derivatives over a region to values on its boundary.

• Green’s theorem — circulation in 2D = double integral of curl over the enclosed region.
• Stokes’ theorem — generalizes Green’s to 3D surfaces.
• Divergence theorem — flux out of a closed surface = volume integral of divergence; Gauss’s theorem.

Complex functions

Functions $f:\mathbb{C}\to \mathbb{C}$ and the topology they live in.

• Complex function — $w=f(z)$; mapping of the plane to itself.
• Limit of complex function — independent of direction of approach.
• Continuity of complex function — limit equals value.

Analytic functions and Cauchy-Riemann

Where complex analysis diverges from real calculus: complex differentiability is enormously restrictive.

• Complex derivative — the limit, when it exists independent of direction.
• Analytic function — complex-differentiable on an open set; equivalent to being represented by a convergent power series.
• Cauchy-Riemann equations — $u_x=v_y$, $u_y=-v_x$; the PDE form of analyticity.
• Harmonic conjugate — the partner $v$ of a harmonic $u$ that makes $u+jv$ analytic.
• Conformality — analytic with nonzero derivative ⇒ angle-preserving.

Elementary complex functions

The standard functions extended to the complex plane, plus their new behaviors.

• Complex exponential — $e^z$; periodic with period $2\pi j$.
• Complex sine and cosine — defined via the exponential; unbounded on $\mathbb{C}$.
• Complex hyperbolic functions — related to sine and cosine by an imaginary shift.
• Complex logarithm — multi-valued; the principal branch and the branch cut.
• Complex power — $z^{\alpha }=e^{\alpha \log z}$; also branch-dependent.

Contour integration

Integrating a complex function along a curve in $\mathbb{C}$.

• Contour integral — ${\int }_{C}f(z)dz$; the complex line integral.
• ML estimate — bound $|{\int }_{C}fdz|\le ML$; the workhorse for showing integrals vanish.

Cauchy’s theorem and integral formula

The core results that make complex analysis what it is.

• Cauchy’s theorem — closed-contour integral of an analytic function is zero.
• Deformation principle — contours can be deformed through analytic regions without changing the integral.
• Cauchy integral formula — value of an analytic function inside a contour is determined by its boundary values: $f(z_0)=\frac{1}{2\pi i}\oint \frac{f(z)}{z-z_0}dz$.
• Cauchy’s estimate — bounds on derivatives from the integral formula.
• Liouville’s theorem — bounded entire functions are constant.
• Maximum modulus principle — $|f|$ takes its maximum on the boundary.
• Winding number — how many times a contour wraps around a point.

Series representations

Local expansions of analytic functions.

• Power series — $\sum a_n(z-z_0{)}^n$ and its radius of convergence.
• Taylor series — power series of an analytic function around a regular point.
• Laurent series — generalized expansion that allows negative powers, valid in an annulus around a singularity.

Singularities and residues

The classification of where analyticity fails, and how to integrate around it.

• Isolated singularity — removable, pole, or essential.
• Residue (complex analysis) — the $a_{-1}$ Laurent coefficient.
• Residue theorem — closed contour integral = $2\pi i$ times sum of enclosed residues; the practical engine of contour integration.
• Jordan’s lemma — bounds for integrals on large semicircles; used to evaluate real Fourier-type integrals by closing in the complex plane.

Conformal mapping

Using analytic maps as geometric transformations.

• Möbius transformation — $w=(az+b)/(cz+d)$; the rational-linear maps.
• Cross-ratio — the Möbius-invariant of four points.
• Smith chart — the Möbius transformation used in RF engineering for impedance matching.
• Riemann sphere — the one-point compactification of $\mathbb{C}$; where Möbius maps are bijections.

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The math here underwrites everything analytical in Differential equations and Signals and systems. The Laplace transform and Inverse Laplace transform are contour integrals — pole locations and residues are the inverse transform. Fourier transform inversion is the same machinery applied to vertical contours. Phasor analysis is just $e^{j\omega t}$ from Euler’s formula applied to steady-state AC circuits. The Smith chart used in transmission-line work is a Möbius transformation in disguise. Laplace’s equation from vector calculus is the equilibrium limit of the diffusion and wave PDEs that show up in fields and electromagnetics.