He also introduced cyclic cohomology in the early 1980s as a first step in the study of noncommutative differential geometry.

Connes has applied his work in areas of mathematics and theoretical physics, including number theory, differential geometry and particle physics.

This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

Other significant results were on Pontryagin duality and differential geometry.

Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry.

Uses synthetic differential geometry and nilpotent infinitesimals.

Johann Carl Friedrich Gauss (;, ) ( 30 April 177723 February 1855 ) was a German mathematician and physical scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.

The survey of Hanover fueled Gauss's interest in differential geometry, a field of mathematics dealing with curves and surfaces.

In differential geometry, one can attach to every point x of a smooth ( or differentiable ) manifold a vector space called the cotangent space at x.

The k-th exterior power of the cotangent space, denoted Λ < sup > k </ sup >( T < sub > x </ sub >< sup >*</ sup > M ), is another important object in differential geometry.

* Conjugate points, in differential geometry

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.

* Hilbert's theorem ( differential geometry )

It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

For a list of differential topology topics, see the following reference: List of differential geometry topics.

Differential topology and differential geometry are first characterized by their similarity.

In one view, differential topology distinguishes itself from differential geometry by studying primarily those problems which are inherently global.

From the point of view of differential geometry, the coffee cup and the donut are different because it is impossible to rotate the coffee cup in such a way that its configuration matches that of the donut.

The system analysis is carried out in time domain using differential equations, in complex-s domain with Laplace transform or in frequency domain by transforming from the complex-s domain.

If the resulting linear differential equations have constant coefficients one can take their Laplace transform to obtain a transfer function.

In mathematical analysis, the Laplace transform is an isomorphism mapping hard differential equations into easier algebraic equations.

Like the Fourier transform, the Laplace transform is used for solving differential and integral equations.

The Laplace transform is another integral operator and is involved in simplifying the process of solving differential equations.

Oliver Heaviside FRS ( ( 18 May 1850 – 3 February 1925 ) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations ( later found to be equivalent to Laplace transforms ), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis.

He invented the operator method for solving linear differential equations, which resembles current Laplace transform methods ( see inverse Laplace transform, also known as the " Bromwich integral ").

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.

In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.

The Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence (∇·) of the gradient (∇ ƒ ).

As a second-order differential operator, the Laplace operator maps C < sup > k </ sup >- functions to C < sup > k − 2 </ sup >- functions for k ≥ 2.

Dirichlet also studied the first boundary value problem, for the Laplace equation, proving the unicity of the solution ; this type of problem in the theory of partial differential equations was later named the Dirichlet problem after him.

* In applications to the physical sciences, operators such as the Laplace operator play a major role in setting up and solving partial differential equations.

The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions.

The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as heat equation, wave equation, Laplace equation and Helmholtz equation.

which are a sequence of linear differential equations that can be readily solved with, for instance, Laplace transform, or Integrating factor.

The radial Laplace operator is split in two differential operators

The first differential operator of the Laplace operator yields

The total Laplace operator yields after applying the second differential operator

In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator.

* Laplace transform is similar to Fourier transform and interchanges operators of multiplication by polynomials with constant coefficient linear differential operators.

Using the Laplace transform, letting, and applying to the differential equation we get

Typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case ( especially in computer implementations ).