13 Glossary
Smooth Manifold: -
Locally: -
Homeomorphic: A description of two spaces that have the same topological structure.
Genus
Holes: An informal synonym for genus.
Tori: A surface that looks like a donut. It has one hole.
Topological Structure: -
Continuous Function: -
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:
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: -
Smooth Function: -
k-Covector: An alternating \(k\)-linear function on a vector space, also known as a multicovector of degree \(k\).
k-Covector Field = k-Form: A function assigning a \(k\)-covector to each point of a manifold, also called a differential \(k\)-form.
k-Tensor: A multilinear function from the \(k\)-fold product of a vector space to the real numbers.
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:
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 (\(\nabla \cdot \nabla V\)).
Linear Operation A synonym for linear map.
Linear Transformation A synonym for linear map.