Explore BrainMass

Explore BrainMass

    Differential Geometry

    Differential geometry is a field of mathematics which possesses similarities to the study of calculus, but differs in how it applies the techniques of integration and differentiation to more complex, higher dimension problems. Differential geometry can study any geometric shape from standard figures to more complex ones, but primarily studies dimensional manifolds. Additionally, curved planes are an area of focus in differential geometry.

    Uncovering the geometry of space and time is a major application of this mathematical field. For example, general relativity, a theory developed by Albert Einstein, used the concepts of differential geometry to define space and time in terms of the shape they take on, which led to the formation of the concept of gravity.

    Considering that differential geometry allows the user to understand the curvature of space and time, this allows it to be applicable and useful in all types of fields. For instance, geological structures can be described through the techniques of differential geometry, similar to how shapes can be analyzed for computer vision using differential geometry.

    The aspects of geometry, specifically narrowed in on the concept of curvature are at the heart of differential geometry. Although differential geometry is a rather abstract field of study which requires the ability to visualize shapes and curves, it has many applications for today’s society. 

    © BrainMass Inc. brainmass.com June 5, 2020, 2:49 am ad1c9bdddf

    BrainMass Solutions Available for Instant Download

    Proofs of the Fibonacci Sequence

    First part is to find an expression in terms of n, the results of the formula: and prove the expression is correct? Secondly A Fibonacci sequence is the basis for a superfast calculation trick as follows: Turn your back and ask someone to write down any two positive integers (vertically and one below the othe

    Biography of a mathematician

    Hello: I am looking for help with the following. If you can do it I need the name of the mathematician ASAP. Biography of a Mathematician Paper Select one mathematician and write an informative biography. To avoid duplication, sign up for the mathematician you would like to research during Week One. Research the selected mathe

    Differential Geometry - Orientable Tangent Bundles

    I need to prove that the tangent bundle of a differentiable manifold is orientable even if the manifold is not. From class, all I know is that a manifold is orientable if it has an atlas {(Ua,xa)} such that when xa(Ua) intersected with xb(Ub) is non-empty then d(xb^(-1) composed with xa) has a positive determinant.

    Relativity: Differential Geometry

    A particle moves along a parametrized curve given by x(lamda)=cos(lamda), y(lamda)=sin(lamda), z(lamda)=lamda Express the path of the curve in the spherical polar coordinates {r, theta, pheta} where x = rsin(theta)cos(pheta) y=rsin(theta)sin(pheta) z=rcos(theta) so that the metric is ds^2=dr^2+(r^2)d(theta)^2+(r^2)sin

    Differential Geometry, imbedded submanifold.

    Let phi : R^2 --> R be a function given by phi(x,y) = x^3 + xy + y^3 +1 For which points p =(0,0) , p=(1/3,1/3), p =(-1/3,-1/3) is the subset phi^-1 ( phi(p)) an imbedded sub-manifold of R^2. ( This question was given as part of Differential Geometry course, we use Do Carmo's book,Riemannian Geometry) Please

    Inverse Function Theorem, Isomorphisms and Diffeomorphisms

    Inverse Function Theorem. Let M and N be differentiable manifolds, φ : M -> N a differentiable mapping and p Є M such that dφp : TM ?> T(p)N is an isomorphism. Prove that is a local diffeomorphism at p; that is, there are neighborhoods U C M of p and V C N of p(p) such that φ : U ?> V is a diffeomorphism. Hint: Coordin

    Differential Geometry : Chain Rule and Differentiable Mappings

    Chain Rule. Let M, N and Q be differentiable manifolds, and let φ : M ?> N and N ?> Q be differentiable mappings. Prove that .... or simply written ..... [Comment: To familiarize yourself with notations in Differential Geometry try to check the form that the above equality takes when you express the (differentials of th