Einstein Notation Derivative, Each term must contain identica

Einstein Notation Derivative, Each term must contain identical non-repeated indices. Tensor contraction can be seen as a generalization of the trace. Participants express confusion about the definitions and relationships between vectors, covectors, and tensors within this framework. Oct 12, 2007 · Understanding Einstein Notation for Derivatives of Fields ehrenfest Oct 12, 2007 Einstein Einstein notation Notation Oct 12, 2007 #1 ehrenfest 2,001 1 Jul 16, 2025 · In order to express partial derivatives, we must use what Ciro Santilli calls the "partial index partial derivative notation", which refers to variables with indices such as x0, x1, x2, ∂ 0, ∂ 1 and ∂ 2 instead of the usual letters x, y and z. e. View style: Other names: Einstein summation convention, summation notation, summation convention Attachments: examples of Einstein summation notation (Example) by bloftin Cross-references: work, domain, spacetime, system, matrix, powers, Einstein There are 3 references to this object. , repeated indices (one upper and one lower) is summed. The gradient (or gradient vector field) of a scalar function f(x1, x2, x3, …, xn) is denoted ∇f or ∇→f where ∇ (nabla) denotes the vector differential operator, del. How would I go about doing this? I figure it's not just going to be ∂μf =gμνxν ∂ μ f = g μ ν x ν. This is version 1 of Einstein summation notation, born on Einstein summation convention differential Ask Question Asked 11 years, 2 months ago Modified 7 years, 10 months ago The Einstein tensor allows the Einstein field equations to be written in the concise form: where is the cosmological constant and is the Einstein gravitational constant. Jul 16, 2025 · In order to express partial derivatives, we must use what Ciro Santilli calls the "partial index partial derivative notation", which refers to variables with indices such as x0, x1, x2, ∂ 0, ∂ 1 and ∂ 2 instead of the usual letters x, y and z. Einstein summation convention differential Ask Question Asked 11 years, 2 months ago Modified 7 years, 10 months ago Feb 4, 2017 · Partial Derivatives in Einstein Notation Ask Question Asked 8 years, 11 months ago Modified 8 years, 11 months ago Nov 27, 2023 · multivariable-calculus vector-analysis differential-operators index-notation einstein-notation Share Cite edited Nov 27, 2023 at 19:40 The notation convention we will use, the Einstein summation notation, tells us that whenever we have an expression with a repeated index, we implicitly know to sum over that index from 1 to 3, (or from 1 to N where N is the dimensionality of the space we are investigating). Does anyone have an intuitive way to do matrix calculus in index/einstein notation? It seems like there's no consistent way people do matrix calculus in standard matrix notation. The solution involves applying the product rule, resulting in $$\partial^\mu (x_\nu x^\nu) = x^a\partial^\mu x_a + x_b\partial^\mu x^b$$, which simplifies to $$2x_\mu$$. Sep 18, 2011 · This discussion focuses on the application of partial differentiation using Einstein notation within the context of the Euler-Lagrange equation. However, in general relativity, it is found that derivatives which are also tensors must be used. take an arbitrary ~r = a^i + b^j + c^z, take its curl, . Repeated indices are implicitly summed over. I think it stems from the way people define the columns and rows of d (vector)/d (vector) which changes from paper to paper I read. The answer of the linked question says that this is because ∂μ ∂ μ is contravariant; however, I have seen many times ∂μ ∂ μ being used the sign convention. Each index can appear at most twice in any term. There are essentially three rules of Einstein summation notation, namely: 1. Oct 6, 2012 · Suppose I have something like f =gμνxμxν f = g μ ν x μ x ν, where the Einstein summation convention is implied. ∂μ:= gμν∂ν ∂ μ:= g μ ν ∂ ν. Tensor (or index, or indicial, or Einstein) notation has been introduced in the previous pages during the discussions of vectors and matrices. Only repeated, raised and lowered indices are summed over. Now suppose I want to to take the derivative ∂μf = ∂f ∂xμ ∂ μ f = ∂ f ∂ x μ. ) Again we're dealing with components and they multiply as numbers so can commute things around to get, Oct 6, 2012 · 2 Suppose I have something like f =gμνxμxν f = g μ ν x μ x ν, where the Einstein summation convention is implied. For example, V ⊗ V, the tensor product of V with itself, has a basis consisting of tensors of the form eij = ei ⊗ ej. Even in special relativity, the partial derivative is still sufficient to describe such changes. Dec 10, 2018 · The discussion revolves around the Einstein summation convention, focusing on its implications for vector and tensor notation, inner products, and operations involving derivatives. Jan 6, 2012 · Very roughly, the derivative has an index, , as a subscript in the denominator. Objects can have more than two indices, also. where (gjk) is the inverse of the matrix (gjk), defined as (using the Kronecker delta, and Einstein notation for summation) . 5), who later jested to a friend, "I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice" Einstein notation In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. Einstein notation: There's an implicitly written summation over repeated indices. The user, Sam Reid, seeks clarification on differentiating expressions involving multiple indices, particularly when summing over one index while differentiating with respect to another. , corresponding to tangent ba-sis element or components of dual vectors) are below whereas contravariant indices (e. The gradient of f is defined as In Einstein notation, this would be . May 24, 2023 · We define a shorthand notation for the partial derivative ∂μ:= ∂ ∂xμ ∂ μ:= ∂ ∂ x μ. 2. By convention, covariant indices (e. Discussion Character Exploratory The gradient of the function f(x,y) = − (cos2x + cos2y)2 depicted as a projected vector field on the bottom plane. We will use Einstein summation notation, i. The derivatives have some common features including that they are derivatives along integral curves of vector fields. , components of tangent vectors or dual basis elements) are above. Feb 3, 2017 · The discussion focuses on calculating the partial derivative of the inner product in Einstein notation, specifically the expression $$\partial^\mu x^2$$. . For a function, the covariant derivative corresponds to DX(f) = X(f), the usual directional derivative, and for vector elds DXY is the connection. From the explicit form of the Einstein tensor, the Einstein tensor is a nonlinear function of the metric tensor, but is linear in the second partial derivatives of the metric. Indices are lowered (raised) with the metric tensor (inverse metric tensor), respectively. . This page reviews the fundamentals introduced on those pages, while the next page goes into more depth on the usefulness and power of tensor notation. My taks is to derive the expression for third-order partial derivatives. The first item on the May 14, 2017 · I've started intro to tensor calculus and I'm pracicing Einstein's summation convention. E. The notation grad f is also commonly used to represent the gradient. Jan 29, 2026 · Einstein summation is a notational convention for simplifying expressions including summations of vectors, matrices, and general tensors. g. (Note that ijk does depend on position so we don't get derivatives of it. The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality. 3. We can now generalize the notion of covariant derivatives to higher order tensors in the \natural" way, satisfying the following properties: Jan 29, 2026 · The convention was introduced by Einstein (1916, sec. The superscripts work just like subscripts, with a different meaning. Without the derivative notation, that becomes a superscript just as a negative power in the denominator of a fraction becomes a positive power in the numerator. Although the Christoffel symbols are written in the same notation as tensors with index notation, they do not transform like tensors under a change of coordinates. May 25, 2021 · However, the Einstein summation convention is being used. Oct 12, 2007 · Understanding Einstein Notation for Derivatives of Fields ehrenfest Oct 12, 2007 Einstein Einstein notation Notation Oct 12, 2007 #1 ehrenfest 2,001 1 Oct 2, 2019 · Problem Taking a Derivative using Einstein Notation Ask Question Asked 5 years, 11 months ago Modified 5 years, 11 months ago Einstein notation makes it easy to manipulate vectors and prove identities that would otherwise be essentially impossible due to how impractical it is to manipulate an arbitrary component-speci c vector (e. bj1hi, lb9h, lum3u, sctc, d1kxj, salt5j, izej, b0y42w, q4dxzz, r949f,