curl(curl F) = grad(div F) - VF
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
16.5
29
![1110
CHAPTER 16 Vector Calculus
23-29 Prove the identity, assuming that the appropriate partial
derivatives exist and are continuous. If f is a scalar field and F, G
are vector fields, then fF, F G, and FX G are defined by
(ƒF)(x, y, z) = f(x, y, z) F(x, y, z)
(FG)(x, y, z) = F(x, y, z)
G(x, y, z)
(F × G)(x, y, z) =
F(x, y, z)
× G(x, y, z)
23. div(F+G) = div F + div G
24. curl(F + G) = curl F + curl G
25. div(ƒF) = f div F + F · Vƒ
26. curl(ƒF) = f curl F + (Vf) X F
*
27. div(F x G) = G curi F - F curl G
28. div(Vfx Vg) = U
29. curl(curl F) = grad(div F) - V²F
30-32 Letr = xi+yj + zk and r = [r].
30. Verify each identity.
(a) ▼·r = 3jab (1)
(c) V²r³ = 12r
URUS
12p)val 10
31. Verify each identity.
(a) Vr = r/r
(c) ▼(1/r) = -r/r³
SS S
(b) ▼ · (rr) = 4r
(I ty
(b) VX r = 0
(d) V ln r = r/r²
32. If F = r/rº, find div F. Is there a value of p for which
div F = 0?
33. Use Green's Theorem in the form of Equation 13 to prove
Green's first identity:
V²g dA = f ƒ(Vg) · nds -
-
SS
D
Vf. Vg dA
where D and C satisfy the hypotheses of Green's Theorem
and the appropriate partial derivatives of f and g exist and are
continuous. (The quantity Vg n = Dng occurs in the line inte-
.
Exercise 33) to show t
fc Dng ds = 0. Here
C
in Exercise 33.
36. Use Green's first iden
on D, and if f(x, y) =
SSD Vf 2 dA = 0. (As
Exercise 33.)
37. This exercise demons
vector and rotations.
z-axis. The rotation c
where w is the angula
of any point P in B di
rotation. Let r= (x,
(a) By considering th
velocity field of
(b) Show that v =
(c) Show that curl v
38. Maxwell's equation
field H as they vary
and no current can I
div E = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1be8149a-7f70-4e17-bee2-140ac0078ae6%2Fc7f70d70-af24-4e52-b215-fee7107fac58%2Fmnuurab_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1110
CHAPTER 16 Vector Calculus
23-29 Prove the identity, assuming that the appropriate partial
derivatives exist and are continuous. If f is a scalar field and F, G
are vector fields, then fF, F G, and FX G are defined by
(ƒF)(x, y, z) = f(x, y, z) F(x, y, z)
(FG)(x, y, z) = F(x, y, z)
G(x, y, z)
(F × G)(x, y, z) =
F(x, y, z)
× G(x, y, z)
23. div(F+G) = div F + div G
24. curl(F + G) = curl F + curl G
25. div(ƒF) = f div F + F · Vƒ
26. curl(ƒF) = f curl F + (Vf) X F
*
27. div(F x G) = G curi F - F curl G
28. div(Vfx Vg) = U
29. curl(curl F) = grad(div F) - V²F
30-32 Letr = xi+yj + zk and r = [r].
30. Verify each identity.
(a) ▼·r = 3jab (1)
(c) V²r³ = 12r
URUS
12p)val 10
31. Verify each identity.
(a) Vr = r/r
(c) ▼(1/r) = -r/r³
SS S
(b) ▼ · (rr) = 4r
(I ty
(b) VX r = 0
(d) V ln r = r/r²
32. If F = r/rº, find div F. Is there a value of p for which
div F = 0?
33. Use Green's Theorem in the form of Equation 13 to prove
Green's first identity:
V²g dA = f ƒ(Vg) · nds -
-
SS
D
Vf. Vg dA
where D and C satisfy the hypotheses of Green's Theorem
and the appropriate partial derivatives of f and g exist and are
continuous. (The quantity Vg n = Dng occurs in the line inte-
.
Exercise 33) to show t
fc Dng ds = 0. Here
C
in Exercise 33.
36. Use Green's first iden
on D, and if f(x, y) =
SSD Vf 2 dA = 0. (As
Exercise 33.)
37. This exercise demons
vector and rotations.
z-axis. The rotation c
where w is the angula
of any point P in B di
rotation. Let r= (x,
(a) By considering th
velocity field of
(b) Show that v =
(c) Show that curl v
38. Maxwell's equation
field H as they vary
and no current can I
div E = 0
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