🏔️ Calculus III: Multivariable

What you’ll learn

1. The Question That Stops Having One AnswerEstablish the course's organising problem: in more than one variable, 'how fast is f changing?' is incomplete until a direction is named.On a hillside the steepness at a single point is anything from +40% to −40% depending only on which way you face, so the question has no answer as posed — unlike f′(3) on a line, where there is only forwards. Every single-variable idea is about to hit this same wall and be rescued the same way: derivative in a direction, accumulation over a region, circulation around a loop. Because direction is what vectors are for, the subject's answers are vectors rather than numbers.
2. A Function of Two Variables Is a LandscapeGet the object and the right way of seeing it: level curves rather than the 3-D surface.f(x, y) returns a number for each point of a plane, and its graph is a surface — a picture that dies at three inputs, since f(x, y, z) would need four dimensions. Level curves, the sets where f equals a constant, describe the function entirely in two dimensions and keep working in any number: topographic contours, weather isobars, isotherms, equipotentials and indifference curves are all the same device. Tightly packed level curves mean fast change, and the direction of fastest change is always perpendicular to them.
3. Freeze Everything But OneDefine partial derivatives as ordinary one-variable derivatives taken along a slice, and see both what they give and what they miss.Holding y fixed reduces f to a function of one variable, so ∂f/∂x is ordinary calculus performed on the curve where a plane cuts the surface — a road cut through a hill. The limitation is that this only ever answers for two arbitrary compass headings, and the question was about the direction you are actually facing. Clairaut's theorem, that the mixed partials agree whenever they are continuous, encodes the fact that both measure the same twist of the surface, and will return as the reason gradient fields cannot circulate.
4. The Vector That Answers Every Direction At OnceDerive the directional derivative, recognise it as a dot product, and understand the gradient as the object that answers every direction simultaneously.Walking a tiny distance along ⟨cos θ, sin θ⟩ and adding the eastward and northward contributions gives D_u f = (∂f/∂x)cos θ + (∂f/∂y)sin θ — a dot product, so D_u f = ∇f · u. Two partial derivatives therefore contain the answer for infinitely many directions. Because D_u f = |∇f|·cos θ, the rate sweeps sinusoidally as you turn on the spot, peaking uphill and vanishing exactly perpendicular to ∇f — which is a one-line proof that the gradient is perpendicular to level curves.
5. The Gradient Points Uphill, and Industry NoticedRead the gradient's geometry straight off the dot product, and see gradient descent as its direct consequence.Since cos θ cannot exceed 1, ∇f is the direction of steepest ascent and |∇f| is how steep — which is why crowded isobars mean strong wind, blowing across them rather than along. If ∇f points uphill then −∇f points downhill, giving gradient descent: stand somewhere, compute the gradient, step against it, repeat. That three-line procedure is how modern machine learning navigates loss landscapes in a billion dimensions that nobody can draw, and it is purely local — it finds the nearest valley, not the deepest.
6. The Tangent PlaneExtend linear approximation to surfaces, and confront the fact that partial derivatives existing is weaker than differentiability.Zoom in on a smooth surface and it becomes a plane, giving f(x, y) ≈ f(a, b) + f_x·(x − a) + f_y·(y − b) — the one-variable linearisation with a second term. But unlike in one variable, having both partials does not guarantee a tangent plane: a surface can be smooth along both axes and crumpled along a diagonal, and the partials, looking only east and north, never notice. Differentiability is the stronger requirement that the approximation works from every direction.
7. The Point That Cannot Exist on a LineClassify critical points, introduce the saddle as the genuinely new possibility, and see why it dominates in high dimensions.∇f = 0 makes every directional derivative vanish, but flat now admits a fourth case: curving up one way and down a perpendicular way — a mountain pass, the highest point of the road and the lowest point of the ridge. The test D = f_xx·f_yy − (f_xy)² sorts them, and the subtracted twist term shows that twisting creates saddles. In high dimensions a minimum needs every direction to agree while a saddle needs one dissenter, so saddles dominate — and they do not trap gradient descent, because their downward directions are exits.
8. Optimising on a LeashDerive Lagrange multipliers as a pure gradient argument, and read λ as a shadow price.Constrained optimisation asks for the highest point of a path across the landscape, which is almost never the summit. Walking the path, you stop climbing exactly when ∇f has no component along it — so ∇f is perpendicular to the path, as is ∇g since g = 0 is its own level curve, and two perpendiculars to the same curve are parallel: ∇f = λ∇g. On a contour map this is where the path grazes a contour rather than crossing it, and λ is the rate at which the optimum improves if the constraint is relaxed.
9. Accumulation Over a RegionExtend integration from intervals to regions, and locate the real difficulty in describing the region rather than in the calculus.A double integral chops a region into patches, weights each by f, and adds — giving mass from a density, area from f = 1, volume from a height. Fubini reduces it to two ordinary integrations, but the inner limits are functions of the outer variable, because the top and bottom of the region depend on where the strip sits; that is where the geometry enters and where the mistakes are made. The order of integration is a real decision: the same region swept the other way can turn an impossible integral into a two-line one.
10. Changing Coordinates, and the Price of Doing SoUnderstand changes of variable as making the region simple, and the Jacobian as the honest price of stretching.A disc in Cartesian coordinates drags square roots through every line; in polar it is a rectangle with four constant limits, because the coordinates now respect the geometry's symmetry. The price is that a polar patch has sides dr and r·dθ — an angular step covers more ground further out — so dA = r dr dθ, a measurement of stretch rather than a rule to memorise. In general the factor is the Jacobian determinant, the local linearisation's area-scaling factor, which for polar works out to exactly r.
11. A Vector at Every PointIntroduce vector fields and the two questions that will occupy the rest of the course.A vector field attaches an arrow to every point — wind, current, gravity, the electric field — and the gradient is already one, derived from a landscape and pointing uphill everywhere. Two questions can be asked of a field, and they are genuinely different: does it circulate, which a paddle wheel would answer by spinning, and does it spread, which a tiny loop would answer by net outflow. Those become curl and divergence, and they are close to exhaustive — Maxwell's equations are structurally four statements about the divergence and curl of two fields.
12. The Fields That Do Not Care How You Got ThereDefine line integrals, identify conservative fields as gradients, and reveal the Fundamental Theorem hiding inside the result.A line integral totals the field's help along a path, and for most fields it depends on the path — as it should. Conservative fields are the exception: the integral depends only on the endpoints, equivalently every closed loop gives zero, which is precisely what licenses assigning a single number like altitude or potential energy to a place. Conservative fields are exactly the gradients, and then ∫_C ∇f · dr = f(end) − f(start) — the same sentence as ∫_a^b f′ = f(b) − f(a), with the interior irrelevant and the boundary deciding everything.
13. Spin and Spread, Made PreciseTurn circulation and flux into formulas, and connect zero curl back to the symmetry of mixed partials.div F = ∂P/∂x + ∂Q/∂y measures net creation at a point — positive is a tap, negative a drain, and div B = 0 is Maxwell's statement that magnetic field lines never begin or end. curl F = ∂Q/∂x − ∂P/∂y measures net twist, and is a vector in three dimensions because a rotation needs an axis. A conservative field has zero curl because curl(∇f) = f_xy − f_yx, which vanishes by Clairaut's theorem — the chapter-three trivia turning out to be the whole reason.
14. The Boundary KnowsDeliver the payoff: Green's, Stokes' and the divergence theorem are the Fundamental Theorem of Calculus in other shapes.Green's theorem equates circulation around a boundary with total curl inside, and the proof is a cancellation — chop the region into cells, and every interior edge is shared by two neighbours and traversed in opposite directions, so only the boundary survives. Stokes' theorem is the same statement on a curved surface, and the divergence theorem the same for flux out of a solid. Line all five up, including ∫_a^b f′ = f(b) − f(a), and every one says the integral of a derivative over a region equals the original object on its boundary — unified in the twentieth century as ∫_M dω = ∫_∂M ω.

Questions this course answers

Grounded in trusted sources

• James Stewart, Daniel Clegg & Saleem Watson — Calculus: Early Transcendentals (9th ed., 2020), chapters 14–16
• Jerrold E. Marsden & Anthony J. Tromba — Vector Calculus (6th ed., 2012)
• Michael Spivak — Calculus on Manifolds (1965), for the unification of the integral theorems as ∫_M dω = ∫_∂M ω
• Gilbert Strang — Calculus (MIT OpenCourseWare, 3rd ed. 2017)
• Tom M. Apostol — Mathematical Analysis (2nd ed., 1974), for differentiability versus existence of partial derivatives
• The gradient-descent curve in this course is computed by running the stated algorithm on the stated function in the build script; it is arithmetic, not measurement.

Every Wunder lesson is built from real, reputable sources — never invented.

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