Examples of differential geometry in the following topics:
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- Simple shapes can be described by basic geometry objects such as a set of two or more points, a line, a curve, a plane, a plane figure (e.g. square or circle), or a solid figure (e.g. cube or sphere).
- Some, such as plant structures and coastlines, may be so arbitrary as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals.
- In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations, rotations (together also called rigid transformations), and uniform scalings.
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- Differentiation, in essence calculating the rate of change, is important in all quantitative sciences.
- Given a function $y=f(x)$, differentiation is a method for computing the rate at which a dependent output $y$ changes with respect to the change in the independent input $x$.
- Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena.
- Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.
- Give examples of differentiation, or rates of change, being used in a variety of academic disciplines
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- Partial derivatives are used in vector calculus and differential geometry.
- Partial differentiation is the act of choosing one of these lines and finding its slope.
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- The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized.
- For a surface given by a differentiable multivariable function $z=f(x,y)$, the equation of the tangent plane at $(x_0,y_0,z_0)$ is given as:
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- The VSPER theory detremines molecular geometries (linear, trigonal, trigonal bipyramidal, tetrahedral, and octahedral).
- Molecular geometries (linear, trigonal, tetrahedral, trigonal bipyramidal, and octahedral) are determined by the VSEPR theory.
- A table of geometries using the VSEPR theory can facilitate drawing and understanding molecules.
- The table of molecular geometries can be found in the first figure.
- Apply the VSEPR model to determine the geometry of a molecule that contains no lone pairs of electrons on the central atom.
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- The study of kinematics is often referred to as the "geometry of motion."
- To describe motion, kinematics studies the trajectories of points, lines and other geometric objects, as well as their differential properties (such as velocity and acceleration).
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- This is an example of a constant coefficient differential equation.
- ($x$ is the zeroth derivative of $x$. ) The most general linear n-th order constant coefficient differential equation is
- These constant coefficient differential equations have a very special property: they reduce to polynomials for exponential $x$.
- The geometry of the Cartesian and polar representations is summarized in Figure 1.5.
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- The arc length can be found using geometry, but for the sake of this atom, we are going to use integration.
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- Four-dimensional Minkowski space-time is only one of many different possible space-times (geometries) which differ in their metric matrix.
- The relation is specified by the Einstein field equations, a system of partial differential equations.
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- In tetrahedral molecular geometry, a central atom is located at the center of four substituent atoms, which form the corners of a tetrahedron.
- This geometry is widespread, particularly for complexes where the metal has d0 or d10 electron configuration.
- The geometry is prevalent for transition metal complexes with d8 configuration.
- In principle, square planar geometry can be achieved by flattening a tetrahedron.
- Therefore, the crystal field splitting diagram for square planar geometry can be derived from the octahedral diagram.