Determine whether or not the given vector field F is conservative. If so, find a potential function. (a) F(x, y) = (x, y) (b) F(x, y) = (3x²y, x³ (c) F(x, y) = (2xe" + y, x² e +x - 2y) (d) F(x, y) = (siny - y sin x+x, cos x + x cos y + y) (e) F(x, y) = (sin(xy) + xy cos(xy), x² cos(xy)) (f) F(x, y, z) = (x, y, z) - (g) F(x, y, z) = (x + 2, −y - z, x − y) (h) F(x, y, z) = (2xy³, x²z³, 3x²yz²) (i) F(x, y, z) = (3y4z2, 4x³ z², -3x²y²) (j) F(x, y, z) = (2x² + 8xy², 3x³y - 3xy, -4y² z² - 2x³z) (k) F(x, y, z) = (y² cosx+23, -4+ 2y sin x, 3xz² + 2) (1) F(x, y, z) = (4xy - 3x²x² + 1,2x² + 2, −2x³z+3z²) Linear vector fields (a) On R² with coordinates x, y, define a vector field to be linear if it is given by a matrix formula = M + V a = -635+6-2+3+; C M = ax+by+e [cx + dy+f] Find necessary and sufficient conditions for to be conservative. Prove that the conditions you wrote are correct. (b) Repeat for a vector field in R3 (using a 3 × 3 matrix in place of M and a 3 × 1 column vector in place of V. (c) Can you generalize to vector fields in R" for any n? Let p be a real number, and let ♬ be a vector field defined on R" - {0} by the formula F(x) = ||||³Ã (of course |||| == Find a potential function for F (the case p = rately). = V 2 should be treated sepa-
Determine whether or not the given vector field F is conservative. If so, find a potential function. (a) F(x, y) = (x, y) (b) F(x, y) = (3x²y, x³ (c) F(x, y) = (2xe" + y, x² e +x - 2y) (d) F(x, y) = (siny - y sin x+x, cos x + x cos y + y) (e) F(x, y) = (sin(xy) + xy cos(xy), x² cos(xy)) (f) F(x, y, z) = (x, y, z) - (g) F(x, y, z) = (x + 2, −y - z, x − y) (h) F(x, y, z) = (2xy³, x²z³, 3x²yz²) (i) F(x, y, z) = (3y4z2, 4x³ z², -3x²y²) (j) F(x, y, z) = (2x² + 8xy², 3x³y - 3xy, -4y² z² - 2x³z) (k) F(x, y, z) = (y² cosx+23, -4+ 2y sin x, 3xz² + 2) (1) F(x, y, z) = (4xy - 3x²x² + 1,2x² + 2, −2x³z+3z²) Linear vector fields (a) On R² with coordinates x, y, define a vector field to be linear if it is given by a matrix formula = M + V a = -635+6-2+3+; C M = ax+by+e [cx + dy+f] Find necessary and sufficient conditions for to be conservative. Prove that the conditions you wrote are correct. (b) Repeat for a vector field in R3 (using a 3 × 3 matrix in place of M and a 3 × 1 column vector in place of V. (c) Can you generalize to vector fields in R" for any n? Let p be a real number, and let ♬ be a vector field defined on R" - {0} by the formula F(x) = ||||³Ã (of course |||| == Find a potential function for F (the case p = rately). = V 2 should be treated sepa-
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
![Determine whether or not the given vector field F is conservative. If so,
find a potential function.
(a) F(x, y) = (x, y)
(b) F(x, y) = (3x²y, x³
(c) F(x, y) = (2xe" + y, x² e +x - 2y)
(d) F(x, y) = (siny - y sin x+x, cos x + x cos y + y)
(e) F(x, y) = (sin(xy) + xy cos(xy), x² cos(xy))
(f) F(x, y, z) = (x, y, z)
-
(g) F(x, y, z) = (x + 2, −y - z, x − y)
(h) F(x, y, z) = (2xy³, x²z³, 3x²yz²)
(i) F(x, y, z) = (3y4z2, 4x³ z², -3x²y²)
(j) F(x, y, z) = (2x² + 8xy², 3x³y - 3xy, -4y² z² - 2x³z)
(k) F(x, y, z) = (y² cosx+23, -4+ 2y sin x, 3xz² + 2)
(1) F(x, y, z) = (4xy - 3x²x² + 1,2x² + 2, −2x³z+3z²)
Linear vector fields
(a) On R² with coordinates x, y, define a vector field to be linear if it is
given by a matrix formula
= M + V
a
=
-635+6-2+3+;
C
M
=
ax+by+e
[cx + dy+f]
Find necessary and sufficient conditions for to be conservative.
Prove that the conditions you wrote are correct.
(b) Repeat for a vector field in R3 (using a 3 × 3 matrix in place of M
and a 3 × 1 column vector in place of V.
(c) Can you generalize to vector fields in R" for any n?
Let p be a real number, and let ♬ be a vector field defined on R" - {0}
by the formula
F(x) = ||||³Ã (of course ||||
==
Find a potential function for F (the case p =
rately).
=
V
2 should be treated sepa-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1d06251-9cb7-4f37-8e8f-dbbf0e17b22e%2F87e33edf-453b-4392-833b-dfe641304a14%2Fkb6sup_processed.png&w=3840&q=75)
Transcribed Image Text:Determine whether or not the given vector field F is conservative. If so,
find a potential function.
(a) F(x, y) = (x, y)
(b) F(x, y) = (3x²y, x³
(c) F(x, y) = (2xe" + y, x² e +x - 2y)
(d) F(x, y) = (siny - y sin x+x, cos x + x cos y + y)
(e) F(x, y) = (sin(xy) + xy cos(xy), x² cos(xy))
(f) F(x, y, z) = (x, y, z)
-
(g) F(x, y, z) = (x + 2, −y - z, x − y)
(h) F(x, y, z) = (2xy³, x²z³, 3x²yz²)
(i) F(x, y, z) = (3y4z2, 4x³ z², -3x²y²)
(j) F(x, y, z) = (2x² + 8xy², 3x³y - 3xy, -4y² z² - 2x³z)
(k) F(x, y, z) = (y² cosx+23, -4+ 2y sin x, 3xz² + 2)
(1) F(x, y, z) = (4xy - 3x²x² + 1,2x² + 2, −2x³z+3z²)
Linear vector fields
(a) On R² with coordinates x, y, define a vector field to be linear if it is
given by a matrix formula
= M + V
a
=
-635+6-2+3+;
C
M
=
ax+by+e
[cx + dy+f]
Find necessary and sufficient conditions for to be conservative.
Prove that the conditions you wrote are correct.
(b) Repeat for a vector field in R3 (using a 3 × 3 matrix in place of M
and a 3 × 1 column vector in place of V.
(c) Can you generalize to vector fields in R" for any n?
Let p be a real number, and let ♬ be a vector field defined on R" - {0}
by the formula
F(x) = ||||³Ã (of course ||||
==
Find a potential function for F (the case p =
rately).
=
V
2 should be treated sepa-
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