1. (a) Using index notation, prove the identity where A is a general vector field. (b) Using index notation, prove the identity F V. V.VX A = 0, grad g = (39 where F and g are general vector and scalar fields respectively. (c) By setting F= ▼ x A, use the identities in parts (a) and (b) to find an expression for div ((curl A) /g) that involves no second derivatives. = (d) Given that general expressions for grad and curl are given, in cylindrical polar basis, by curl B = eR + V.F-F.Vg g² R - find the curl of the vector field B to evaluate the divergence of the field 1 ag R 06 eo + 2e₂ R+z eR Re ez ə ə ə ƏR аф Əz BR RB B₂ Reo+z²e and hence use the result of part (c) RE-X ag əz 2 ezi
1. (a) Using index notation, prove the identity where A is a general vector field. (b) Using index notation, prove the identity F V. V.VX A = 0, grad g = (39 where F and g are general vector and scalar fields respectively. (c) By setting F= ▼ x A, use the identities in parts (a) and (b) to find an expression for div ((curl A) /g) that involves no second derivatives. = (d) Given that general expressions for grad and curl are given, in cylindrical polar basis, by curl B = eR + V.F-F.Vg g² R - find the curl of the vector field B to evaluate the divergence of the field 1 ag R 06 eo + 2e₂ R+z eR Re ez ə ə ə ƏR аф Əz BR RB B₂ Reo+z²e and hence use the result of part (c) RE-X ag əz 2 ezi
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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Transcribed Image Text:1. (a) Using index notation, prove the identity
where A is a general vector field.
(b) Using index notation, prove the identity
▼.
V. V × A = 0,
F gV.FF. Vg
g²
g
grad g =
where F and g are general vector and scalar fields respectively.
(c) By setting F = V × A, use the identities in parts (a) and (b) to find an expression
for div ((curl A)/g) that involves no second derivatives.
მყ
ᎧᎡ
(d) Given that general expressions for grad and curl are given, in cylindrical polar basis,
by
=
curl B =
er +
R
-
find the curl of the vector field B
to evaluate the divergence of the field
1 ag
R do
(
eR
ə
ƏR
eo +
Re
ə
მი
BR RB B₂
Reo+²ez and hence use the result of part (c)
2e₂
R+z
52-5
ag
əz
ez
ə
əz
ezi
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