(a) You should already be able to prove, by direct computation of the required differ- entiations in Cartesian coordinates, that the curl of a gradient is always zero and the divergence of a curl is always zero. In a similar manner, verify the following identities in which and are arbitrary differentiable scalar functions of position, and F and G are arbitrary differentiable vector functions of position: (i) ▼ · (F) = &V · F + F · V (ii) ▼ × (▼ × F) = V (VF) – V²F - (b) Calculate the divergence of the function: F(x, y, z) = f(x)â + ƒ(y)ŷ + ƒ(−22) and show that it is zero at the point (c, c, -c/2). [Hint: chain-rule.] (c) Calculate the curl of the following function. -yzx+xzy in Cartesian coordinates. Now write the same equation in cylindrical coordinates and calculate its curl in cylindrical coordinates. Convert your answer back to Cartesian coordinates and compare to the answer you obtained previously. (Note: all information required can be found on the inside covers of Griffith's.)
(a) You should already be able to prove, by direct computation of the required differ- entiations in Cartesian coordinates, that the curl of a gradient is always zero and the divergence of a curl is always zero. In a similar manner, verify the following identities in which and are arbitrary differentiable scalar functions of position, and F and G are arbitrary differentiable vector functions of position: (i) ▼ · (F) = &V · F + F · V (ii) ▼ × (▼ × F) = V (VF) – V²F - (b) Calculate the divergence of the function: F(x, y, z) = f(x)â + ƒ(y)ŷ + ƒ(−22) and show that it is zero at the point (c, c, -c/2). [Hint: chain-rule.] (c) Calculate the curl of the following function. -yzx+xzy in Cartesian coordinates. Now write the same equation in cylindrical coordinates and calculate its curl in cylindrical coordinates. Convert your answer back to Cartesian coordinates and compare to the answer you obtained previously. (Note: all information required can be found on the inside covers of Griffith's.)
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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can you help me solve the parts and show workings please
![(a) You should already be able to prove, by direct computation of the required differ-
entiations in Cartesian coordinates, that the curl of a gradient is always zero and
the divergence of a curl is always zero. In a similar manner, verify the following
identities in which and are arbitrary differentiable scalar functions of position,
and F and G are arbitrary differentiable vector functions of position:
(i) ▼ · (F) = &V · F + F · V
(ii) ▼ × (▼ × F) = V (VF) – V²F
-
(b) Calculate the divergence of the function:
F(x, y, z) = f(x)â + ƒ(y)ŷ + ƒ(−22)
and show that it is zero at the point (c, c, -c/2). [Hint: chain-rule.]
(c) Calculate the curl of the following function.
-yzx+xzy
in Cartesian coordinates. Now write the same equation in cylindrical coordinates
and calculate its curl in cylindrical coordinates. Convert your answer back to
Cartesian coordinates and compare to the answer you obtained previously. (Note:
all information required can be found on the inside covers of Griffith's.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6713d75e-2fb3-4074-927a-cea8cea15561%2F1ccc2077-a978-4238-a59d-c7317d772eee%2Fypz11uk_processed.png&w=3840&q=75)
Transcribed Image Text:(a) You should already be able to prove, by direct computation of the required differ-
entiations in Cartesian coordinates, that the curl of a gradient is always zero and
the divergence of a curl is always zero. In a similar manner, verify the following
identities in which and are arbitrary differentiable scalar functions of position,
and F and G are arbitrary differentiable vector functions of position:
(i) ▼ · (F) = &V · F + F · V
(ii) ▼ × (▼ × F) = V (VF) – V²F
-
(b) Calculate the divergence of the function:
F(x, y, z) = f(x)â + ƒ(y)ŷ + ƒ(−22)
and show that it is zero at the point (c, c, -c/2). [Hint: chain-rule.]
(c) Calculate the curl of the following function.
-yzx+xzy
in Cartesian coordinates. Now write the same equation in cylindrical coordinates
and calculate its curl in cylindrical coordinates. Convert your answer back to
Cartesian coordinates and compare to the answer you obtained previously. (Note:
all information required can be found on the inside covers of Griffith's.)
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