mathematics

Rivlin–Ericksen tensor

Rivlin–Ericksen tensor is a mathematics 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 Rivlin–Ericksen tensor rather than just read about it. In short: A Rivlin–Ericksen temporal evolution of the strain rate tensor such that the derivative translates and rotates with the flow field. The first-order Rivlin–Ericksen is given by A i j ( 1 ) = ∂ v i ∂ x j + ∂ v j ∂ x i {\displaystyle \mathbf {A} _{ij(1)}={\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}} where v i {\displaystyle v_{i}} is the fluid's velocity and A i j ( n ) {\displaysty…

Key takeaways

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

Reference excerpt

A Rivlin–Ericksen temporal evolution of the strain rate tensor such that the derivative translates and rotates with the flow field. The first-order Rivlin–Ericksen is given by

A i j ( 1 ) = ∂ v i ∂ x j + ∂ v j ∂ x i {\displaystyle \mathbf {A} _{ij(1)}={\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}}

where

v i {\displaystyle v_{i}} is the fluid's velocity and

A i j ( n ) {\displaystyle A_{ij(n)}} is n {\displaystyle n} -th order Rivlin–Ericksen tensor. Higher-order tensor may be found iteratively by the expression

A i j ( n + 1 ) = D D t A i j ( n ) . {\displaystyle A_{ij(n+1)}={\frac {\mathcal {D}}{{\mathcal {D}}t}}A_{ij(n)}.}

The derivative chosen for this expression depends on convention. The upper-convected time derivative, lower-convected time derivative, and Jaumann derivative are often used.

References Truesdell, Clifford & Noll, Walter (2004). The Non-Linear Field Theories of Mechanics. Springer. ISBN 978-3-662-10388-3.

Worked examples

Example 1 — a first encounter with Rivlin–Ericksen tensor

Start with the simplest possible case. Write down what Rivlin–Ericksen tensor claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Rivlin–Ericksen tensor 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 Rivlin–Ericksen tensor 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 Rivlin–Ericksen tensor

In research: Rivlin–Ericksen tensor appears in mathematics 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 Rivlin–Ericksen tensor 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: Rivlin–Ericksen tensor is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fluid dynamics, Multivariable calculus, Non-Newtonian fluids, so understanding it makes those chapters shorter.
In everyday life: Look for Rivlin–Ericksen tensor 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 Rivlin–Ericksen tensor in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rivlin–Ericksen tensor 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 Rivlin–Ericksen tensor out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rivlin–Ericksen tensor in simple terms?

A Rivlin–Ericksen temporal evolution of the strain rate tensor such that the derivative translates and rotates with the flow field. The first-order Rivlin–Ericksen is given by A i j ( 1 ) = ∂ v i ∂ x j + ∂ v j ∂ x i {\displaystyle \mathbf {A} _{ij(1)}={\frac {\partial v_{i}}{\partial x_{j}}}+{\frac…

Why does Rivlin–Ericksen tensor matter?

Because it connects several mathematics 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 Rivlin–Ericksen tensor?

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 Rivlin–Ericksen tensor.