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In particular, a C k-atlas that is C 0-compatible with a C 0-atlas that defines a topological manifold is said to determine a C k differential structure on the topological manifold. The C k equivalence classes of such atlases are the distinct C k differential structures of the manifold. Each distinct differential structure is determined by a ...
The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus. Dynamic equations on a time scale have a potential for applications such as in population dynamics .
The mapping φ is called an immersion if its differential is injective at every point of U. The image of φ is called an immersed submanifold. More specifically, for m = 3, which means that the ambient Euclidean space is ℝ 3, the induced metric tensor is called the first fundamental form. Suppose that φ is an immersion onto the submanifold M ...
A topological manifold that is in the image of is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold". Thus given two categories, the two natural questions are:
Maps whose rank is generically maximal, but drops at certain singular points, occur frequently in coordinate systems. For example, in spherical coordinates, the rank of the map from the two angles to a point on the sphere (formally, a map T 2 → S 2 from the torus to the sphere) is 2 at regular points, but is only 1 at the north and south ...
Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
The Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero. This is the inverse function theorem.
In R 3, the gradient, curl, and divergence are special cases of the exterior derivative. An intuitive interpretation of the gradient is that it points "up": in other words, it points in the direction of fastest increase of the function. It can be used to calculate directional derivatives of scalar functions or normal directions. Divergence ...