19 Glossary
Smooth Manifold: A topological manifold equipped with a maximal smooth atlas.
Locally: A description that every point has a neighborhood homeomorphic to a Euclidean open set.
Homeomorphic: A description of two spaces that have the same topological structure.
Genus: The maximum number of cuts along nonintersecting closed loops possible without disconnecting a surface.
Holes: An informal synonym for genus.
Tori: A surface that looks like a donut. It has one hole.
Topological Structure: A collection of subsets, called open sets, closed under arbitrary unions and finite intersections.
Continuous Function: A function whose limit as it approaches any number equals its value there.
Differential Form: A function assigning an alternating k-linear function to each point of an open set or manifold.
Surface: A 2-dimensional manifold, an object modeled locally on \(\mathbb{R}^2\), specifying points with two independent parameters.
Piecewise Smooth: A property of curves or surfaces composed of finitely many smooth segments meeting at corners or edges.
Oriented Surface: A surface with a consistent choice of orientation at each point, typically specified by a nowhere-vanishing 2-form.
Smooth Surface: A 2-dimensional manifold equipped with a differentiable structure, allowing for calculus and well-defined tangent spaces.
Surface Normal: A vector perpendicular to the tangent plane of a surface.
Piecewise Smooth Simple Closed Curve: A non-self-intersecting closed path consisting of finitely many differentiable segments joined end-to-end.
Open Set: A set that is open in a topological space.
Open Set (Expanded): A subset of a topological space that contains a neighborhood around every one of its points.
Differentiable Function: A mapping that can be locally approximated at a point by a linear transformation.
Smooth Function: A function that is infinitely differentiable in all combinations of its variables.
Alternating Function: A function that changes sign whenever any two of its arguments are switched.
k-Covector: An alternating function of \(k\) vectors that is linear in each vector.
k-Covector Field = k-Form: An assignment of a \(k\)-covector to each point of a manifold.
k-Tensor: A function of \(k\) vectors that is linear in each vector.
k-Fold: A term denoting the repetition of a set or space \(k\) times, typically within a Cartesian product.
k-Linear: A synonym for multilinear function.
k-Fold Product: The Cartesian product of \(k\) copies of a set \(X\), often denoted simply as \(X^k\).
Bijection: A function that is both injective and surjective, also referred to as a one-to-one correspondence.
Hausdorff Space: A topological space where any two distinct points can be separated by disjoint open neighborhoods.
Onto: A synonym for surjective function.
One-to-One: A synonym for an injective function.
Covariant Vector = Covector = 1-Form: A linear, real-valued function of a single vector input.
Gradient: \(\nabla V\) Operating on a scalar function \(V\) produces a vector field.
Divergence: \(\nabla \cdot \mathbf{A}\) The dot product of the operator with a vector field \(\mathbf{A}\) yields a scalar field.
Curl: \(\nabla \times \mathbf{A}\) The cross product of the operator with a vector field \(\mathbf{A}\) results in another vector field.
Laplacian: \(\nabla^2 V\) This operator represents the divergence of the gradient (\(\nabla \cdot \nabla V\)).
Legendre Transform: A mathematical operation that changes independent variables to construct a new function.
Linear Map = Linear Transformation: A mapping that preserves addition of vectors and multiplication by scalars that is used to transform vectors from one space to another.
Linear Operation A synonym for linear map.
Hermitian: A linear transformation that is used to represent physical quantities with real number eigenvalues.