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In Hungarian Rhapsody No. 6, Franz Liszt takes the unusual step of changing the key from D-flat major to C-sharp major near the start of the piece, and then back again to B-flat minor. Maurice Ravel selected C-sharp major as the tonic key of "Ondine" from his piano suite Gaspard de la nuit.
A major theorem, often called the fundamental theorem of the differential geometry of surfaces, asserts that whenever two objects satisfy the Gauss-Codazzi constraints, they will arise as the first and second fundamental forms of a regular surface. Using the first fundamental form, it is possible to define new objects on a regular surface.
Flat and sharp are mutually inverse isomorphisms of smooth vector bundles, hence, for each p in M, there are mutually inverse vector space isomorphisms between T p M and T ∗ p M. The flat and sharp maps can be applied to vector fields and covector fields by applying them to each point. Hence, if X is a vector field and ω is a covector field,
Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces.
The field of differential geometry became an area of study considered in its own right, distinct from the more broad idea of analytic geometry, in the 1800s, primarily through the foundational work of Carl Friedrich Gauss and Bernhard Riemann, and also in the important contributions of Nikolai Lobachevsky on hyperbolic geometry and non ...
Vector field; Tensor field; Differential form; Exterior derivative; Lie derivative; pullback (differential geometry) pushforward (differential) jet (mathematics) Contact (mathematics) jet bundle; Frobenius theorem (differential topology) Integral curve
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.
The metric tensor (,) induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat ♭ and sharp ♯. A section A ∈ Γ ( T M ) {\displaystyle A\in \Gamma (TM)} corresponds to the unique one-form A ♭ ∈ Ω 1 ( M ) {\displaystyle A^{\flat }\in \Omega ^{1}(M)} such that for all sections X ∈ Γ ( T M ...