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Differential geometry finds applications throughout mathematics and the natural sciences. Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity, and subsequently by physicists in the development of quantum field theory and the standard model of particle physics.
See also multivariable calculus, list of multivariable calculus topics. Manifold. Differentiable manifold; Smooth manifold; Banach manifold; Fréchet manifold; Tensor analysis. Tangent vector
A vector field X on M and a vector field Y on N are said to be φ-related if φ ∗ X = φ ∗ Y as vector fields along φ. In other words, for all x in M, dφ x (X) = Y φ(x). In some situations, given a X vector field on M, there is a unique vector field Y on N which is φ-related to X. This is true in particular when φ is a diffeomorphism.
That is, df is the unique 1-form such that for every smooth vector field X, df (X) = d X f , where d X f is the directional derivative of f in the direction of X. The exterior product of differential forms (denoted with the same symbol ∧) is defined as their pointwise exterior product.
In differential calculus, the domain-straightening theorem states that, given a vector field on a manifold, there exist local coordinates , …, such that = / in a neighborhood of a point where is nonzero.
A section of a tangent vector bundle is a vector field. A vector bundle E {\displaystyle E} over a base M {\displaystyle M} with section s {\displaystyle s} . In the mathematical field of topology , a section (or cross section ) [ 1 ] of a fiber bundle E {\displaystyle E} is a continuous right inverse of the projection function π ...
Let M be a Banach manifold of class C r with r ≥ 2. As usual, TM denotes the tangent bundle of M with its natural projection π M : TM → M given by : (,). A vector field on M is a cross-section of the tangent bundle TM, i.e. an assignment to every point of the manifold M of a tangent vector to M at that point.
Given any curve c : (a, b) → S, one may consider the composition X ∘ c : (a, b) → ℝ 3. As a map between Euclidean spaces, it can be differentiated at any input value to get an element (X ∘ c)′(t) of ℝ 3. The orthogonal projection of this vector onto T c(t) S defines the covariant derivative ∇ c ′(t) X.