Suppose that f is a function of two variables (y and z) only. Show that the gradient Vƒ = (@ƒ/ây)ŷ + (âƒ/əz)2 transforms as a vector un- der rotations, Eq. 1.29. [Hint: (âƒ/əy) = (əƒ/ây)(Əy/Əy) + (Əƒ/əz)(əz/Əy), and the analogous formula for af/az. We know that y = y cos+ z sino and Z = -y sin + z cosp; "solve" these equations for y and z (as functions of y and 7), and compute the needed derivatives ay/ay, az/ay, etc.]
Suppose that f is a function of two variables (y and z) only. Show that the gradient Vƒ = (@ƒ/ây)ŷ + (âƒ/əz)2 transforms as a vector un- der rotations, Eq. 1.29. [Hint: (âƒ/əy) = (əƒ/ây)(Əy/Əy) + (Əƒ/əz)(əz/Əy), and the analogous formula for af/az. We know that y = y cos+ z sino and Z = -y sin + z cosp; "solve" these equations for y and z (as functions of y and 7), and compute the needed derivatives ay/ay, az/ay, etc.]
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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Please help answer the question (see image for question and necessary equation).
![Suppose that f is a function of two variables (y and z) only.
Show that the gradient Vƒ = (@ƒ/ây)ŷ + (âƒ/əz)2 transforms as a vector un-
der rotations, Eq. 1.29. [Hint: (âƒ/Əy) = (əƒ/ây)(Əy/Əy) + (Əƒ/əz)(əz/Əy),
and the analogous formula for af/az. We know that y = y cos + z sino and
Z = -y sin + z cosp; "solve" these equations for y and z (as functions of y
and 7), and compute the needed derivatives ay/ay, az/ay, etc.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0a3387dc-1b17-43e3-a267-a6713aa700b8%2F3bd83020-e3ab-44af-8775-4fc404bb4321%2Fjsuouzc_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose that f is a function of two variables (y and z) only.
Show that the gradient Vƒ = (@ƒ/ây)ŷ + (âƒ/əz)2 transforms as a vector un-
der rotations, Eq. 1.29. [Hint: (âƒ/Əy) = (əƒ/ây)(Əy/Əy) + (Əƒ/əz)(əz/Əy),
and the analogous formula for af/az. We know that y = y cos + z sino and
Z = -y sin + z cosp; "solve" these equations for y and z (as functions of y
and 7), and compute the needed derivatives ay/ay, az/ay, etc.]
Transcribed Image Text:(3) - (
Ay
=
Az
cos p
sin
-
-) (A₂
Ay
Az
sin o
cos
(1.29)
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