Multiple integrals extend integration to dimensions of two and more, and with them arrive a change of variables that no elementary analogue suggests, a family of coordinate systems tailored to particular symmetries, and the vector calculus in which integrals over volumes, surfaces and lines become the natural way to state physical conservation laws. An iterated integral is a limit of sums over a rectangle in a higher dimensional space, and its value does not depend on the order in which the integrations are performed provided the integrand is absolutely integrable, a fact known as Fubini's theorem which rests on the same measure-theoretic foundations as the one dimensional case. Regions that are awkward in Cartesian coordinates often turn out to be simple after a change of variables, and the transformation law carries an extra factor: the volume element is multiplied by the absolute value of the Jacobian determinant, the rate at which the transformation stretches volume locally, and for a linear map this determinant is simply the ratio of the volumes of the stretched to the original parallelepiped. Polar coordinates in the plane replace the rectangular area element by a radial element times an angular element times the radius, which explains both why a disc of radius a has area equal to pi times a squared and why the area of a circular annulus grows in proportion to its circumference. Cylindrical and spherical coordinates do the same job in three dimensions, turning integrals over a solid of revolution or a sphere into one dimensional calculations, and the spherical case additionally makes the volume element shrink to zero along a ray, a fact of geometric rather than mystical interest. The applications of this machinery are immediate and physical. Volumes of solids with curved boundaries, masses and centres of mass of bodies with nonuniform density, moments of inertia that determine how hard a flywheel is to spin up, gravitational fields of the Earth approximated by an oblate ellipsoid, and the total charge on an irregular conductor are all computed as triple integrals. In probability the same formulas express the joint density of two random variables, the expected value of a function of two independent measurements, and the covariance that measures their linear dependence, while normalisation conditions demand that the integral of the density over the whole plane equal one. Numerical practice has its own dialect. Simple rules such as the midpoint and trapezoid rules achieve errors proportional to the square of the step size, Simpson's rule gains a further power by fitting a parabola, and Gaussian quadrature reaches high accuracy with few evaluations by choosing nodes and weights that are optimal rather than equally spaced. In higher dimensions the curse of dimensionality appears: a uniform grid refined in each direction needs the number of nodes raised to the dimension, so that a problem of modest accuracy becomes unaffordable in ten coordinates, and Monte Carlo estimation sidesteps this by sampling randomly and averaging, which reduces the error as the inverse square root of the number of samples rather than a power of it. Importance sampling and quasi-Monte Carlo sequences, in which low discrepancy point sets replace independent draws, improve the effective rate, and both techniques appear in rendering, reliability analysis and global optimisation. Vector calculus generalises derivatives to fields, and the two quantities at its centre are the divergence and the curl. The divergence of a vector field is the net outflow per unit volume at a point, computed as the sum of the partial derivatives of its components, and a field whose divergence vanishes everywhere carries no sources: fluid that enters a region must also leave it, and a magnetic field with zero divergence is one with no monopoles. The curl measures local rotation, combining the derivatives of the components in crossed pairs, and a field with zero curl is conservative, meaning that a potential function exists whose gradient is the field, so that the line integral of the field around any closed curve vanishes. The circulation form of the curl is the more useful one for computation, since it can be evaluated as a line integral around small loops, and this is how fluid velocities and magnetic intensities are actually measured. Three theorems tie these notions to integration and are collectively known as the generalised fundamental theorem of the calculus. Green's theorem turns a line integral around a closed curve in the plane into a double integral of the curl over the enclosed region, and it powers the proof that a field with zero curl has a potential and gives the plane form of the circulation of a vortex. Stokes' theorem generalises it to surfaces, equating the surface integral of the curl to the line integral around the boundary, which is the natural statement that a magnetic field cannot be created by a current loop confined to its own path. The divergence theorem, sometimes called Gauss's theorem, converts the flux of a field through a closed surface into the integral of its divergence over the enclosed volume, and in fluid mechanics and electromagnetism it expresses conservation laws, such as the incompressibility of water, or the statement that a flux of field lines through a closed surface measures the enclosed source. Together with a healthy supply of orientation signs these theorems are indispensable in fluid dynamics, where they appear in the derivation of the Navier-Stokes equations, in electromagnetic field theory, in the divergence and curl forms of Maxwell's equations, and in the numerical methods that discretise those equations on a mesh.
