Course · Electrical engineering track

Calculus III for Engineers

Multivariable and vector calculus: the geometry of space, curves and motion, partial derivatives and optimisation, multiple integrals, vector fields, and the three big theorems that turn into Maxwell’s equations. Every unit is pointed at the place an electrical engineer actually meets it. Gradients exist so you can find the direction an electric field pushes. Flux integrals exist so you can write Gauss’s law. Stokes’ theorem exists so Faraday’s law makes sense. The lessons that need spatial intuition — surfaces, volumes, fields — come with interactive labs built right into the page.

Units

Unit 1 · 3 labs

Vectors & the Geometry of Space

3D coordinates, dot and cross products, lines, planes, quadric surfaces, cylindrical and spherical coordinates.

Unit 2 · 1 lab

Vector Functions & Motion in Space

Space curves, derivatives, arc length, the TNB frame, curvature, velocity and acceleration, charged particles in fields.

Unit 3 · 2 labs

Partial Derivatives

Surfaces and contour maps, partials, tangent planes, chain rule, the gradient, directional derivatives, extrema, Lagrange multipliers.

Unit 4 · 3 labs

Multiple Integrals

Double and triple integrals, changing the order, polar, cylindrical and spherical, Jacobians, mass and moments.

Unit 5 · 2 labs

Vector Fields & Line Integrals

Fields, work and circulation, conservative fields and potentials, Green’s theorem, divergence and curl.

Unit 6 · 2 labs

Surface Integrals, Stokes & Gauss

Parametric surfaces, flux, Stokes’ theorem, the divergence theorem — and Maxwell’s equations as the payoff.

Practice exams

One per unit, ten questions each, ~90 minutes, full worked solutions hidden behind a click so you can actually test yourself. Roughly two thirds pure technique and one third applied to fields and circuits — close to the balance a real engineering midterm uses.

Exam 1

Vectors & geometry

10 questions · 90 min

Exam 2

Vector functions

10 questions · 90 min

Exam 3

Partial derivatives

10 questions · 90 min

Exam 4

Multiple integrals

10 questions · 90 min

Exam 5

Vector fields & line integrals

10 questions · 90 min

Exam 6

Surface integrals & theorems

10 questions · 90 min

What you need coming in

Calculus I and II, and specifically these six things. If any is shaky, patch it now — every one of them shows up in week one.

The derivative and antiderivative tables cold. Partial derivatives are ordinary derivatives with the other variables frozen; there is no new differentiation to learn, only new bookkeeping.
The chain rule, fluently. Unit 3 generalises it to trees of dependencies and the whole of Unit 5 depends on it.
Integration techniques — substitution above all, plus parts. Multiple integrals are single integrals nested; the inner one still has to be done.
Polar coordinates from Calculus II. Cylindrical is polar with a \(z\) bolted on; spherical is the 3D version.
Trig identities and the unit circle, including where \(\cos\) and \(\sin\) are positive. Spherical coordinates and every rotational field lean on this.
Basic vectors — components, magnitude, dot product. Unit 1 rebuilds all of it from scratch, but the 3D vectors lab is the fastest refresher.

Coordinate systems and their volume elements

Every integral in this course is written in one of three coordinate systems, and the thing people forget is not the coordinates — it is the stretch factor that goes with them. Memorise this table; Unit 4 derives it, Units 5 and 6 use it constantly.

\(\theta\) is the same angle in cylindrical and spherical — measured in the \(xy\)-plane from the positive \(x\)-axis. \(\phi\) is measured *down from the positive \(z\)-axis*, so the equator is \(\phi = \pi/2\). Physics and EE texts swap the letters \(\theta\) and \(\phi\); check which convention your course uses.
System	Coordinates	Conversion to Cartesian	Volume element \(dV\)	Natural for
Cartesian	\((x, y, z)\)	—	\(dx\,dy\,dz\)	Boxes, planes, anything with flat faces
Cylindrical	\((r, \theta, z)\), \(r \ge 0\)	\(x = r\cos\theta,\; y = r\sin\theta,\; z = z\)	\(r\,dr\,d\theta\,dz\)	Cylinders, cones, paraboloids, coaxial cable, anything symmetric about the \(z\)-axis
Spherical	\((\rho, \theta, \phi)\), \(\rho \ge 0\), \(0 \le \phi \le \pi\)	\(x = \rho\sin\phi\cos\theta,\; y = \rho\sin\phi\sin\theta,\; z = \rho\cos\phi\)	\(\rho^2\sin\phi\,d\rho\,d\theta\,d\phi\)	Spheres, point charges, antenna radiation patterns, anything symmetric about a point

The del operator, four ways

One symbol, \(\nabla = \langle \partial_x, \partial_y, \partial_z\rangle\), and the four things it can do. Units 3, 5 and 6 introduce them one at a time; here they are side by side so you can see the pattern.

Operation	Acts on	Gives	Formula (Cartesian)	Meaning
Gradient \(\nabla f\)	scalar field	vector field	\(\langle f_x, f_y, f_z\rangle\)	Direction and rate of steepest increase. \(\mathbf{E} = -\nabla V\).
Divergence \(\nabla\cdot\mathbf{F}\)	vector field	scalar field	\(\dfrac{\partial F_1}{\partial x} + \dfrac{\partial F_2}{\partial y} + \dfrac{\partial F_3}{\partial z}\)	Net outflow per unit volume — source strength. \(\nabla\cdot\mathbf{E} = \rho/\varepsilon_0\).
Curl \(\nabla\times\mathbf{F}\)	vector field	vector field	\(\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ \partial_x&\partial_y&\partial_z\\ F_1&F_2&F_3\end{vmatrix}\)	Rotation per unit area — how a paddle wheel would spin. \(\nabla\times\mathbf{E} = -\partial_t\mathbf{B}\).
Laplacian \(\nabla^2 f\)	scalar field	scalar field	\(f_{xx} + f_{yy} + f_{zz}\)	Divergence of the gradient. \(\nabla^2 V = -\rho/\varepsilon_0\) (Poisson); \(\nabla^2 V = 0\) in empty space (Laplace).

The two identities that decide everything

\(\nabla\times(\nabla f) = \mathbf{0}\). The curl of a gradient is always zero. So a field with nonzero curl cannot be a gradient — it has no potential, and its line integrals depend on the path. This is why an induced electric field has no voltage function.
\(\nabla\cdot(\nabla\times\mathbf{F}) = 0\). The divergence of a curl is always zero. So a curl field has no sources — which is exactly the statement that magnetic field lines never begin or end.

Why this course matters for EE specifically

Calculus III is where the mathematics of fields lives. Electromagnetics, antenna theory, semiconductor physics and control theory are all written in this language. Here is the map, so you know what you are buying with each unit:

Unit	Where it lands in the EE curriculum
1 — Vectors & geometry	The cross product is the Lorentz force. \(\mathbf{F} = q\mathbf{v}\times\mathbf{B}\), torque on a motor coil, the Poynting vector \(\mathbf{E}\times\mathbf{H}\). Cylindrical coordinates are coaxial cables; spherical are point charges and antennas.
2 — Vector functions	Charged-particle trajectories in E and B fields (cyclotron and helical motion), and the parametrised paths every line integral in Unit 5 runs along.
3 — Partial derivatives	\(\mathbf{E} = -\nabla V\). The gradient turns a potential map into a field. Small-signal transistor models are multivariable linearisation; tolerance analysis is the total differential; Lagrange multipliers are constrained circuit optimisation.
4 — Multiple integrals	Total charge is \(\iiint\rho\,dV\); stored field energy is \(\iiint \tfrac12\varepsilon E^2\,dV\); capacitance and inductance of real geometries come from triple integrals in cylindrical and spherical coordinates. Joint probability densities in noise analysis are double integrals.
5 — Vector fields & line integrals	Voltage is a line integral of \(\mathbf{E}\). Path independence is Kirchhoff’s voltage law. Ampère’s law is a circulation. Curl and divergence are the differential forms of Maxwell.
6 — Surface integrals & theorems	Flux is Gauss’s law. The divergence theorem converts it to \(\nabla\cdot\mathbf{E} = \rho/\varepsilon_0\); Stokes’ theorem converts Faraday’s and Ampère’s laws to their differential forms. This unit ends with all four Maxwell equations in both forms.

How to work through this

Read the unit page and play with every lab — drag the view, move the sliders, break the presets. The labs are there because these ideas are three-dimensional and a static picture cannot show you what a saddle point or a flux integral is really doing. Then close the page and do the practice exam cold, timed, no notes. For every miss, go back to the section it came from rather than just reading the solution. Each exam question maps to exactly one section of the unit page.