Glossary

The operator \partial that sends a chain to the chain of its oriented edges -- and satisfies \partial(\partial c)=0 always.

A point where every ball, no matter how small, catches points both inside and outside the set.

A multidimensional array whose entries transform in a specific way under a rotation of axes.

A formal sum of oriented cells (points, edges, triangles, ...) -- the domain a differential form is integrated over.

How much a vector field pushes along a closed loop, all the way around -- the line integral of the field around that loop.

A form \omega with d\omega=0.

A set that contains all of its own boundary points.

Closed AND bounded -- the property that guarantees an absolute max and min exist.

A quantity stays constant along every path the system can actually follow, rather than changing.

\partial\rho/\partial t + \nabla\cdot(\rho\mathbf{v}) = 0 -- the local statement that a conserved quantity is neither created nor destroyed.

Getting arbitrarily close to a fixed value as a process continues -- the terms of a sequence or series settle down instead of wandering or blowing up.

Any coordinate system built from a smooth map, not necessarily straight axes -- polar, cylindrical, and spherical are the familiar cases.

The measurement of exactly how many closed forms on a region fail to be exact -- a number that detects holes.

A single number computed from a square matrix that tells whether it's invertible, and by how much it stretches or flips space.

Has a well-defined derivative at a point -- the function is smooth enough there for a tangent line or plane to exist.

An object built to be integrated -- a 1-form eats a vector, a 2-form eats a pair, and so on.

The number of independent directions or coordinates needed to describe a space or object.

A scalar measuring how much a vector field spreads outward (positive) or converges inward (negative) at a point.

The necessary condition for y to extremize \int L(x,y,y')\,dx: \partial L/\partial y - d/dx(\partial L/\partial y')=0.

The operator d that turns a k-form into a (k{+}1)-form -- grad, curl, and div are all just d in disguise.

A machine that eats an entire function and returns a single number -- arc length, energy, or travel time, for example.

The factorial, generalized to non-integer inputs: \Gamma(n+1)=n! for integers, defined for all real n>0 by an integral.

The shortest path between two points on a given surface -- a straight line in flat space, a great-circle arc on a sphere.

The vector of a function's partial derivatives -- it points the direction the function grows fastest, and its length is that fastest growth rate.

The fundamental solution encoding how a point source's effect spreads out -- superpose it to solve for any source distribution.

A function satisfying Laplace's equation, \nabla^2 u = 0 -- no interior local max or min.

Every sufficiently nice vector field splits into a curl-free part (a gradient) plus a divergence-free part (a curl).

The matrix of all second partial derivatives -- the quadratic term in a function's Taylor expansion.

When \partial F/\partial y \neq 0, the level set F(x,y)=0 can be solved for y as a function of x, at least locally.

If Df(a) is invertible, f itself is locally invertible near a -- and the inverse is just as smooth.

A map or matrix that can be exactly undone -- there's a second map that reverses every step of the first.

The matrix of all first partial derivatives -- the total derivative written out in coordinates.

The Riemann-integration notion of "volume" -- a set has Jordan measure zero if it can be boxed by rectangles of arbitrarily small total volume.

A differential form of degree k -- built to be integrated over a k-dimensional oriented piece.

\delta_{ij}=1 if i=j, else 0 -- the index-notation stand-in for the identity matrix.

\varepsilon_{ijk}=\pm1 for an even/odd permutation of 1,2,3, else 0 -- the index-notation cross product.

The rate of change felt by a moving fluid parcel, D/Dt = \partial/\partial t + \mathbf{v}\cdot\nabla -- not the rate of change at a fixed point.

The higher-dimensional generalization of a disk (2D) or solid ball (3D) -- all points within a fixed radius in n dimensions.

Every continuous symmetry of a functional's integrand produces a corresponding conserved quantity along its extremals.

A set where every point has some wiggle room -- a small enough ball around it still fits inside.

A consistent choice of "which way is positive" across a curve, surface, or space -- for example, which side counts as the outward-facing side.

Meeting at a right angle -- for vectors, this means their dot product is zero.

The physical length of a unit step in one curvilinear coordinate -- h_i = \|\partial\mathbf{r}/\partial u_i\|.

A region with no holes -- every loop drawn inside it can be shrunk down to a point without leaving the region.

Unchanged when flipped across its diagonal (a matrix), or when its inputs are swapped (a product or form).

The best linear approximation to f at a point -- a single matrix, Df(\mathbf{a}), not a list of separate partials.

The curl of a velocity field, \boldsymbol\omega=\nabla\times\mathbf{v} -- twice the local angular velocity of the fluid.

The antisymmetric product that builds higher-degree forms: \alpha\wedge\beta = -\beta\wedge\alpha.