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b) V·(v.)= v·(V.) + :Vy (where T denotes transpose) 1 c) Dy oy O² = -+-V(y·v)-y×(Vxv) Dt where D ə +v.V is known as the "substantial derivative" operator. Dt ôt

b) V·(v.)= v·(V.) + :Vy (where T denotes transpose) 1 c) Dy oy O² = -+-V(y·v)-y×(Vxv) Dt where D ə +v.V is known as the "substantial derivative" operator. Dt ôt

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need B and C please!

Prove the following identities using simplified index notation: (Vu) at b) T V·(y•t)= y·(V• 7² ) + :Vy (where 7 denotes transpose) Dv ὃν Dt where = + = V(v • v) — v × (V×v) Ət D Dt Ə -+v. V is known as the "substantial derivative" operator. Ət Hint: for part (c), it will be easier to work from the right side toward the left.
Transcribed Image Text:Prove the following identities using simplified index notation: (Vu) at b) T V·(y•t)= y·(V• 7² ) + :Vy (where 7 denotes transpose) Dv ὃν Dt where = + = V(v • v) — v × (V×v) Ət D Dt Ə -+v. V is known as the "substantial derivative" operator. Ət Hint: for part (c), it will be easier to work from the right side toward the left.
Expert Solution
Introduction

In this solution, we will derive two important vector identities.

First, we will derive an expression for the divergence of a tensor product of a vector and a second-order tensor.

Second, we will derive an expression for the substantial derivative of a vector, which is a key concept in the study of fluid motion.

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