Calculus III

A partial derivative measures how a multivariable function changes as ONE variable changes, holding all others fixed. For a surface z = f(x,y), the partial derivative with respect to x is the slope of the curve you get by slicing the surface with a plane of constant y.

The directional derivative D_u f generalizes the partial derivative: instead of measuring change along just the x-axis or y-axis, it measures the rate of change of f in an arbitrary direction u. Partial derivatives ∂f/∂x and ∂f/∂y are just the special cases where u = (1,0) or u = (0,1).

The gradient ∇f of a multivariable function f collects all its partial derivatives into a single vector. That vector has a striking geometric meaning: it points in the direction of steepest increase of f at a given point, and its magnitude tells you exactly how steep that climb is. The gradient is the multivariable generalization of the derivative and is the single most important object in multivariable optimization, from Lagrange multipliers to gradient descent in machine learning.

Lagrange multipliers solve constrained optimization problems: find the maximum or minimum of f(x,y) subject to a constraint g(x,y)=c. Rather than solving the constraint for one variable and substituting (often messy or impossible), the method exploits a beautiful geometric fact — at a constrained extremum, the gradients of f and g must be parallel.

A double integral ∬_D f(x,y) dA extends the single-variable integral to compute volume under a surface z=f(x,y) over a 2D region D, by slicing the solid into infinitesimal boxes instead of rectangles. Triple integrals ∭_E f(x,y,z) dV extend this further to sum a quantity over a 3D solid region. Multiple integrals are evaluated in practice using Fubini's theorem, which reduces them to iterated single-variable integrals, and their difficulty often hinges on choosing the right coordinate system (Cartesian, polar, cylindrical, spherical).

A multivariable function takes more than one input variable and produces an output — most commonly f(x,y) or f(x,y,z). Instead of a curve, the graph of a two-variable function z=f(x,y) is a surface in 3D space. Multivariable functions are how calculus models anything depending on multiple independent quantities at once: temperature depending on position, profit depending on price and quantity, or a neural network's loss depending on thousands of weights.

Cartesian coordinates (x, y, z) describe every point in space with three perpendicular distances, but many solids — cylinders, cones, spheres, planetary orbits — have symmetry that Cartesian coordinates hide and complicate. Cylindrical coordinates (r, θ, z) replace the x and y axes with polar coordinates in the horizontal plane while keeping z unchanged, ideal for objects with an axis of rotational symmetry. Spherical coordinates (ρ, θ, φ) instead describe a point by its distance ρ from the origin, an azimuthal angle θ around the z-axis, and a polar angle φ down from the positive z-axis — ideal for objects with symmetry about a single point, like spheres and radiating fields. Both systems trade some of Cartesian coordinates' simplicity for equations that become dramatically simpler once the right symmetry is exploited.

The single-variable chain rule computes d/dt[f(g(t))] = f'(g(t))·g'(t) — the rate of change of a composed function is the product of the rates of change along the way. The multivariable chain rule generalizes this when a function z = f(x,y) depends on x and y, and x and y in turn depend on other variables (like time t, or two other parameters u and v). Instead of a single product, every path from the output back to the input variable contributes a term, and the total rate of change is the SUM over all such paths. This 'multivariable chain rule' underlies related rates problems in several variables, implicit differentiation of multivariable equations, and backpropagation in neural networks (where a loss depends on outputs, which depend on weights, through many layered compositions).

Some regions — disks, annuli, circular sectors, cardioid-shaped regions — are far more naturally described using polar coordinates (r, θ) than Cartesian (x, y). Converting a double integral over such a region into polar coordinates replaces awkward Cartesian bounds (often involving square roots) with clean bounds on r and θ, at the cost of an extra factor of r appearing in the area element: dA = r dr dθ. This factor is not optional bookkeeping — it is the Jacobian of the polar transformation, accounting for the fact that a small 'rectangle' of sides dr and dθ in polar coordinates actually sweeps out a physical area of r dr dθ, growing with distance from the origin.

Just as a tangent LINE gives the best linear approximation to a single-variable function at a point, a tangent PLANE gives the best linear approximation to a surface z = f(x,y) at a point. The tangent plane touches the surface at one point and matches both partial derivatives there, hugging the surface closely nearby. This lets us replace a complicated surface with a simple flat plane for small movements away from the point of tangency — the linear approximation (or linearization) L(x,y), which underlies error estimation, optimization algorithms, and the very definition of differentiability in several variables.

Divergence and curl are two scalar/vector operators that describe the local behavior of a vector field. Divergence div F = ∇·F is a scalar measuring how much a field 'spreads out' (sources) or 'converges' (sinks) at a point — how much flux flows out of an infinitesimal volume. Curl curl F = ∇×F is a vector measuring the field's local rotation or 'spin' at a point — the axis and strength of infinitesimal circulation. Together they are the local, differential versions of the flux and circulation integrals that Green's, Stokes', and the divergence theorems relate to global (integral) behavior.

The divergence theorem (also called Gauss's theorem or Ostrogradsky's theorem) relates the flux of a vector field out of a closed surface S to the triple integral of the field's divergence over the solid region E that S encloses. It's the 3D analogue of Green's theorem, converting a (often hard) surface flux calculation into a (often easier) volume integral, or vice versa.

Green's theorem relates a line integral around a simple closed curve C (traversed counterclockwise) to a double integral over the region D that C encloses. It converts a potentially hard 1D circulation calculation into an easier 2D area calculation, or vice versa, and is the 2D special case of the more general Stokes' theorem.

A line integral generalizes ordinary integration from a straight interval [a,b] to an arbitrary curve C in the plane or in space. A scalar line integral ∫_C f ds sums a scalar quantity (like mass density) along the curve's arc length; a vector line integral ∫_C F·dr sums the component of a vector field F along the curve's direction of travel, computing quantities like work done by a force along a path.

Stokes' theorem generalizes Green's theorem from a flat region in the plane to a curved surface in 3D space: it relates the circulation of a vector field around the boundary curve ∂S of an oriented surface S to the flux of the field's curl through S itself. It's the 3D statement that 'total spin on the boundary equals total local rotation summed over the surface.'

A surface integral extends the double integral from a flat 2D region to a curved surface in 3D space. A scalar surface integral ∬_S f dS sums a quantity (like mass density) over a surface's area; a vector (flux) surface integral ∬_S F·dS measures how much a vector field flows through the surface — the total flux, essential for the divergence theorem and Stokes' theorem.

A vector field assigns a vector — not just a number — to every point in space. Where a scalar field like temperature T(x,y,z) gives a single number at each point, a vector field like wind velocity or a magnetic field gives both a magnitude and a direction at each point. Vector fields are the natural setting for describing flows, forces, and fluxes, and are the objects that line integrals, surface integrals, divergence, and curl all operate on.

A space curve traced by a vector-valued function r(t) = (x(t), y(t), z(t)) has a length that can be computed by integrating the speed |r'(t)| over the parameter interval — the natural 3D generalization of the arc-length formula from single-variable calculus. Beyond length, curvature κ measures how sharply a curve bends at each point: a straight line has curvature 0 everywhere, while a tight circle has large constant curvature. Together, arc length and curvature let us describe a curve's shape independent of how fast it happens to be traversed, forming the basis of the Frenet-Serret frame used in the differential geometry of curves.