A partial differential equation PDE is a differential equation that contains unknown multivariable functions and their partial derivatives. This is in contrast to ordinary differential equations , which deal with functions of a single variable and their derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena in nature such as sound , heat , electrostatics , electrodynamics , fluid flow , elasticity , or quantum mechanics.

These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems , partial differential equations often model multidimensional systems. PDEs find their generalization in stochastic partial differential equations.

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A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives the linearity or non-linearity in the arguments of the function are not considered here. There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries.

Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory cf.

Navier—Stokes existence and smoothness. However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution. Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations see below.

Differential equations are described by their order, determined by the term with the highest derivatives.

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An equation containing only first derivatives is a first-order differential equation , an equation containing the second derivative is a second-order differential equation , and so on. Two broad classifications of both ordinary and partial differential equations consists of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones. In the next group of examples, the unknown function u depends on two variables x and t or x and y.

Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest. For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. The solution may not be unique. See Ordinary differential equation for other results.

However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:.

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The theory of differential equations is closely related to the theory of difference equations , in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

The study of differential equations is a wide field in pure and applied mathematics , physics , and engineering. All of these disciplines are concerned with the properties of differential equations of various types.

Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.

Instead, solutions can be approximated using numerical methods. Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics , differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations.

Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. The derivative of the basis vector is a vector, thus it can be written as a linear combination of the basis vectors:. Differentiating a one form is done using the fact, that is a scalar, thus.

This is obviously a tensor, because the above equation has a tensor on the left hand side and tensors on the right hand side and. Similarly for the derivative of the tensor we use the fact that is a vector:. If the vectors at infinitesimally close points of the curve are parallel and of equal length, then is said to be parallel transported along the curve, i.

In components using the tangent vector :. We require orthogonality , in a general frame:. This is called a Thomas precession. For any vector, we define: the vector is Fermi-Walker tranported along the curve if:. If is perpendicular to , the second term is zero and the result is called a Fermi transport. Why: the is transported by Fermi-Walker and also this is the equation for gyroscopes, so the natural, nonrotating tetrade is the one with , which is then correctly transported along any curve not just geodesics. Geodesics is a curve that locally looks like a line, i.

So we can see that the equation is invariant as long as , which gives:. Another way to derive the geodesic equation is by finding a curve that extremizes the proper time:. Here can be any parametrization. We have introduced to make the formulas shorter:. We vary this action with respect to :. By setting the variation we obtain the geodesic equation:. We have a freedom of choosing , so we choose such parametrization so that , which makes and we recover 3. Note that the equation 3.

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## Differential Geometry — Theoretical Physics Reference documentation

Curvature means that we take a vector , parallel transport it around a closed loop which is just applying a commutator of the covariant derivatives and see how it changes. We express the result in terms of the vector :. The coefficients form a tensor called Riemann curvature tensor. Expanding the left hand side:. Where we have used the fact that all terms symmetric in in particular and and get canceled by the same term in the. We get. Using these expressions for the curvature tensor in a local inertial frame, we derive the following 5 symmetries of the curvature tensor by simply substituting for the left hand side and verify that it is equal to the right hand side:.

These are tensor expressions and so even though we derived them in a local inertial frame, they hold in all coordinates. The last identity is called a Bianchi identity. From the last equality we can see that it is symmetric in. A Ricci scalar is defined as:. It is symmetric in due to the symmetry of the metric and Ricci tensors. By contracting the Bianchi identity twice, we can show that Einstein tensor has zero divergence:. Definition of the Lie derivative of any tensor is:. In general, the Christoffel symbols are not symmetric and there is no metric that generates them. However, if the manifold is equipped with metrics, then the fundamental theorem of Riemannian geometry states that there is a unique Levi-Civita connection, for which the metric tensor is preserved by parallel transport:.

We define the commutation coefficients of the basis by. In general these coefficients are not zero as an example, take the units vectors in spherical or cylindrical coordinates , but for coordinate bases they are. It can be proven, that. All last 3 expressions are used but the last one is probably the most common. At the beginning we used the usual trick that is symmetric but is antisymmetric. Later we used the identity , which follows from the well-known identity by substituting and taking the logarithm of both sides.

I was reading a book on Riemannian analysis and the author assumes some formulas of differential geometry , which may be basic but I have a lack of knowledge on those.

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I have never seen those rules and it is not the first time in this book or another that I'm blocked reading a calculation proof because of a lack of formulas. I tried to find a "cheat sheet" of formulas but didn't find more than one about the stoke formula but nothing that encompasses rules such as the ones needed above. Any good pointer or "must read" to not fall short of common formulas? I'm also fine with ad hoc rules for the 2 bullet points mentioned above. Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. What formulas of differential geometry am I missing?