physics

Second covariant derivative

Second covariant derivative is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Second covariant derivative rather than just read about it. In short: In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields. Definition Formally, given a (pseudo)-Riemannian manifold (M, g) associated with a vector bundle E → M, let ∇ denote the Levi-Civita connection given by the metric g, and d…

Key takeaways

Second covariant derivative belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Second covariant derivative to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Second covariant derivative from memory before moving on to harder problems.

Reference excerpt

In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields.

Definition Formally, given a (pseudo)-Riemannian manifold (M, g) associated with a vector bundle E → M, let ∇ denote the Levi-Civita connection given by the metric g, and denote by Γ(E) the space of the smooth sections of the total space E. Denote by T*M the cotangent bundle of M. Then the second covariant derivative can be defined as the composition of the two ∇s as follows:

Γ ( E ) ⟶ ∇ Γ ( T ∗ M ⊗ E ) ⟶ ∇ Γ ( T ∗ M ⊗ T ∗ M ⊗ E ) . {\displaystyle \Gamma (E){\stackrel {\nabla }{\longrightarrow }}\Gamma (T^{*}M\otimes E){\stackrel {\nabla }{\longrightarrow }}\Gamma (T^{*}M\otimes T^{*}M\otimes E).}

For example, given vector fields u, v, w, a second covariant derivative can be written as

( ∇ u , v 2 w ) a = u c v b ∇ c ∇ b w a {\displaystyle (\nabla _{u,v}^{2}w)^{a}=u^{c}v^{b}\nabla _{c}\nabla _{b}w^{a}}

by using abstract index notation. It is also straightforward to verify that

( ∇ u ∇ v w ) a = u c ∇ c v b ∇ b w a = u c v b ∇ c ∇ b w a + ( u c ∇ c v b ) ∇ b w a = ( ∇ u , v 2 w ) a + ( ∇ ∇ u v w ) a . {\displaystyle (\nabla _{u}\nabla _{v}w)^{a}=u^{c}\nabla _{c}v^{b}\nabla _{b}w^{a}=u^{c}v^{b}\nabla _{c}\nabla _{b}w^{a}+(u^{c}\nabla _{c}v^{b})\nabla _{b}w^{a}=(\nabla _{u,v}^{2}w)^{a}+(\nabla _{\nabla _{u}v}w)^{a}.}

Thus

∇ u , v 2 w = ∇ u ∇ v w − ∇ ∇ u v w . {\displaystyle \nabla _{u,v}^{2}w=\nabla _{u}\nabla _{v}w-\nabla _{\nabla _{u}v}w.}

When the torsion tensor is zero, so that [ u , v ] = ∇ u v − ∇ v u {\displaystyle [u,v]=\nabla _{u}v-\nabla _{v}u} , we may use this fact to write Riemann curvature tensor as

R ( u , v ) w = ∇ u , v 2 w − ∇ v , u 2 w . {\displaystyle R(u,v)w=\nabla _{u,v}^{2}w-\nabla _{v,u}^{2}w.}

Similarly, one may also obtain the second covariant derivative of a function f as

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Second covariant derivative

Start with the simplest possible case. Write down what Second covariant derivative claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Second covariant derivative before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Second covariant derivative ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Second covariant derivative

In research: Second covariant derivative appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Second covariant derivative in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Second covariant derivative is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematical physics stubs, Relativity stubs, Riemannian geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Second covariant derivative outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Second covariant derivative in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Second covariant derivative means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Second covariant derivative out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Second covariant derivative in simple terms?

Why does Second covariant derivative matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Second covariant derivative?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Second covariant derivative.