(b) Recall that a level set of g: R³ → R is defined as Sc = {(x, y, z) E R³: g(x, y, z) = C} for some constant C. Consider the specific function g(x, y, z) = ex²+2y² +3z². Show that the level sets of g are ellipsoids for C> 1. What are the level sets for C<1? (c) Let C = e. Find the equation for the level set Se. Show that the curve y(t) = (x(t), y(t), z(t)) = - (₁. V. √(1-P) √2 t2 3 is contained in Se for all t€ [-]. Then show that y(t) · Vg(y(t)) = 0 for all t.
(b) Recall that a level set of g: R³ → R is defined as Sc = {(x, y, z) E R³: g(x, y, z) = C} for some constant C. Consider the specific function g(x, y, z) = ex²+2y² +3z². Show that the level sets of g are ellipsoids for C> 1. What are the level sets for C<1? (c) Let C = e. Find the equation for the level set Se. Show that the curve y(t) = (x(t), y(t), z(t)) = - (₁. V. √(1-P) √2 t2 3 is contained in Se for all t€ [-]. Then show that y(t) · Vg(y(t)) = 0 for all t.
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
![(a) Consider a function f: R³ → R defined by f(x, y, z) = x² + ey- z. Show that the
directional derivative Daf(x, y, z) has maximum magnitude when û = f(x,y,z)
That is, show that Vf does in fact give the direction of greatest change.
Vf(x,y,z)
(b) Recall that a level set of g: R³ R is defined as
Sc = {(x, y, z) € R³: g(x, y, z) = C}
for some constant C. Consider the specific function g(x, y, z) = ex²+2y²+3z². Show
that the level sets of g are ellipsoids for C> 1. What are the level sets for C≤ 1?
(c) Let C = e. Find the equation for the level set Se. Show that the curve
y(t) = (x(t), y(t), z(t)) (1, √21, √/13 - 12
is contained in Se for all t € [33]. Then show that y(t) · Vg(y(t)) = 0 for
√3'
all t.
1
=
Q Search
H
DELL
zoom](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2F8d081810-5100-422b-aaa1-ce3d7d5faf0b%2F3o4f05_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Consider a function f: R³ → R defined by f(x, y, z) = x² + ey- z. Show that the
directional derivative Daf(x, y, z) has maximum magnitude when û = f(x,y,z)
That is, show that Vf does in fact give the direction of greatest change.
Vf(x,y,z)
(b) Recall that a level set of g: R³ R is defined as
Sc = {(x, y, z) € R³: g(x, y, z) = C}
for some constant C. Consider the specific function g(x, y, z) = ex²+2y²+3z². Show
that the level sets of g are ellipsoids for C> 1. What are the level sets for C≤ 1?
(c) Let C = e. Find the equation for the level set Se. Show that the curve
y(t) = (x(t), y(t), z(t)) (1, √21, √/13 - 12
is contained in Se for all t € [33]. Then show that y(t) · Vg(y(t)) = 0 for
√3'
all t.
1
=
Q Search
H
DELL
zoom
Expert Solution
Step 1
(b)
The given set is for some constant .
The function is defined as ,
(c) Given , The level set is , and the given curve is
for
The gradient of a function is defined as .
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