→Problem 1: Assume that the vector field E = (E₁, E2, E3) : R³ R³ is twice continu-ously differentiable (E; € C²(R³), i = 1, 2, 3). Show thatV x (V x E) = V(V · E) — V · VE.Derivatives are taken componentwise, for example,V · VE = (V · VE₁, V VE2, V · VE3).

Given that,The vector field is double differentiable.Here, is known as the gradient operator and…

Answered: Problem 1: Assume that the vector field…

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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Problem 1: Assume that the vector field E = (E₁, E2, E3) : R³ R³ is twice continu-
ously differentiable (E; € C²(R³), i = 1, 2, 3). Show that
V x (V x E) = V(V · E) — V · VE.
Derivatives are taken componentwise, for example,
V · VE = (V · VE₁, V VE2, V · VE3).

Expert Solution

Step 1: Introduction

Given that,

The vector field $E space equals open parentheses E subscript 1 comma space E subscript 2 comma space E subscript 3 close parentheses space colon space R cubed rightwards arrow with blank on top R cubed$ is double differentiable.

Here, $nabla equals i with hat on top fraction numerator partial differential over denominator partial differential x end fraction plus j with hat on top fraction numerator partial differential over denominator partial differential y end fraction plus k with hat on top fraction numerator partial differential over denominator partial differential z end fraction$ is known as the gradient operator and $cross times text and end text space times$ are the symbols of cross and dot product respectively.

Now the cross product $nabla cross times E$ is given as:

$nabla cross times E equals open vertical bar table row cell i with hat on top end cell cell j with hat on top end cell cell k with hat on top end cell row cell fraction numerator partial differential over denominator partial differential x end fraction end cell cell fraction numerator partial differential over denominator partial differential y end fraction end cell cell fraction numerator partial differential over denominator partial differential z end fraction end cell row cell E subscript 1 end cell cell E subscript 2 end cell cell E subscript 3 end cell end table close vertical bar nabla cross times E equals i with hat on top open parentheses fraction numerator partial differential E subscript 3 over denominator partial differential y end fraction minus fraction numerator partial differential E subscript 2 over denominator partial differential z end fraction close parentheses minus j with hat on top open parentheses fraction numerator partial differential E subscript 3 over denominator partial differential x end fraction minus fraction numerator partial differential E subscript 1 over denominator partial differential z end fraction close parentheses plus j with hat on top open parentheses fraction numerator partial differential E subscript 2 over denominator partial differential x end fraction minus fraction numerator partial differential E subscript 1 over denominator partial differential y end fraction close parentheses$