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Calculate ▽ × F for the following vector fields: (a) F (r, 2y,32) (c) F (32,y2,.2),...
. Calculate .F for the tollowing vector ficlds: (a) F (r, 2y,3z) (c) F (32,y2,22) (b)...
. Calculate .F for the tollowing vector ficlds: (a) F (r, 2y,3z) (c) F (32,y2,22) (b) F- (3a
3. Consider the vector field F(x, y) + 2y F dr, where C is the circle (r-2)2 +y2 = 1, oriented counterclock (a) Com...
3. Consider the vector field F(x, y) + 2y F dr, where C is the circle (r-2)2 +y2 = 1, oriented counterclock (a) Compute wise (Hint: use the FT of line integrals. We could not use it for the circle centered at the origin, but we can use the theorem for this circle. Why?) (b) Let 0 be the angle in polar coordinates for a point (x, y). Check that 0 is a potential function for F
3. Consider the...
Question 2 For the following vector fields F determine whether or not they are conservative. For ...
Only the Matlab part !!!
Question 2 For the following vector fields F determine whether or not they are conservative. For the conservative vector fields, construct a potential field f (i.e. a scalar field f with Vf - F) (a) F(z, y)(ryy,) (b) F(z, y)-(e-y, y-z) (c) F(r, y,z) (ry.y -2, 22-) (d) F(x, y, z)=(-, sin(zz),2, y-rsin(x:) Provide both your "by hand" calculations alongside the MATLAB output to show your tests for the whether they are conservative, and to...
Determine if the following vector fields F: 2 CR" + R" are conservative. In case they...
Determine if the following vector fields F: 2 CR" + R" are conservative. In case they are conservative, find a potential function f, that is, such that F= Vf. a) F(1, y) = (x²y, zy), N=R? b) F(1, y, z) = (ze", 22 sin(z), 2+z+1), N=R3 c) F(x,y) = (e cosy, -efsiny), R=R2
(c) Let F be the vector field on R given by F(x, y, z) = (2x...
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
2. Compute | F. ds for each of the vector fields F and paths r given...
2. Compute | F. ds for each of the vector fields F and paths r given below: (b) Ple:) - (a ) and re) – () witte (0.1 Fler,1,2) = ( and r(t) = ( ) with t e (0, 2). F(x, y, z) = | 22 and r(t) = with t€ (0,2). F(x, y, z) = sin Cos y 32 and r(t) = -t with t € (0,1). (a) F(x, y, z) = | Vies:)-( .) --( * )-464...
6. Calculate the following line integrals of vector fields. Be sure to name any theorems you use;...
6. Calculate the following line integrals of vector fields. Be sure to name any theorems you use; if you don't use a theorem, write "calculated directly2 (d) F . dr, where F(x,y)-(2ry-уг, r2 +3y2-2cy), and C is the piecewise-linear path frorn (1,3) to (5,2 to (12) to (4,1) (e) φ F.dr, where F(z,y)-(3ysin(Zy), 3rsin(2y)+6ry cos(2p)), and C is the ellipse 2 +9y2-64. oriented counter-clockwise
6. Calculate the following line integrals of vector fields. Be sure to name any theorems you...
For each of the following vector fields, find its curl and determine if it is a...
For each of the following vector fields, find its curl and
determine if it is a gradient field.
(1 point) For each of the following vector fields, find its curl and determine if it is a gradient field. (a) F = 5(xy + 22) + 10(x2 + y2) 7+ 10(x2 + y2) k. curl F = F ? (b) Ğ = 5yzi + (52z+z2) 7+ (5xy + 2yz) k: curl Ĝ = Ğ ? (c) H = (5xy + yz)...
(1 point) (a) Show that each of the vector fields F-4yi + 4x j, G-i ЗУ x2+y2 x?+yi J, and j are g...
(1 point) (a) Show that each of the vector fields F-4yi + 4x j, G-i ЗУ x2+y2 x?+yi J, and j are gradient vector fields on some domain (not necessarily the whole plane) x2+y2 by finding a potential function for each. For F, a potential function is f(x, y) - For G, a potential function is g(x, y) - For H, a potential function is h(x, y) (b) Find the line integrals of F, G, H around the curve C...
c. Let F : R³ → R³ be a vector field on R, given by the...
c. Let F : R³ → R³ be a vector field on R, given by the following function F(x, y, 2) = (x2)i + (y2)J + (xy)k. Calculate the flux of the field across the surface of the hemisphere, : [0, 1] × [0, 2x] → R³, where parametrized by the following function Þ(r, 0) = (r cos 0)i + (r sin 0) + (1 – 1²)!/2 k.
. Calculate .F for the tollowing vector ficlds: (a) F (r, 2y,3z) (c) F (32,y2,22) (b) F- (3a
3. Consider the vector field F(x, y) + 2y F dr, where C is the circle (r-2)2 +y2 = 1, oriented counterclock (a) Compute wise (Hint: use the FT of line integrals. We could not use it for the circle centered at the origin, but we can use the theorem for this circle. Why?) (b) Let 0 be the angle in polar coordinates for a point (x, y). Check that 0 is a potential function for F
3. Consider the...
Only the Matlab part !!!
Question 2 For the following vector fields F determine whether or not they are conservative. For the conservative vector fields, construct a potential field f (i.e. a scalar field f with Vf - F) (a) F(z, y)(ryy,) (b) F(z, y)-(e-y, y-z) (c) F(r, y,z) (ry.y -2, 22-) (d) F(x, y, z)=(-, sin(zz),2, y-rsin(x:) Provide both your "by hand" calculations alongside the MATLAB output to show your tests for the whether they are conservative, and to...
Determine if the following vector fields F: 2 CR" + R" are conservative. In case they are conservative, find a potential function f, that is, such that F= Vf. a) F(1, y) = (x²y, zy), N=R? b) F(1, y, z) = (ze", 22 sin(z), 2+z+1), N=R3 c) F(x,y) = (e cosy, -efsiny), R=R2
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
2. Compute | F. ds for each of the vector fields F and paths r given below: (b) Ple:) - (a ) and re) – () witte (0.1 Fler,1,2) = ( and r(t) = ( ) with t e (0, 2). F(x, y, z) = | 22 and r(t) = with t€ (0,2). F(x, y, z) = sin Cos y 32 and r(t) = -t with t € (0,1). (a) F(x, y, z) = | Vies:)-( .) --( * )-464...
6. Calculate the following line integrals of vector fields. Be sure to name any theorems you use; if you don't use a theorem, write "calculated directly2 (d) F . dr, where F(x,y)-(2ry-уг, r2 +3y2-2cy), and C is the piecewise-linear path frorn (1,3) to (5,2 to (12) to (4,1) (e) φ F.dr, where F(z,y)-(3ysin(Zy), 3rsin(2y)+6ry cos(2p)), and C is the ellipse 2 +9y2-64. oriented counter-clockwise
6. Calculate the following line integrals of vector fields. Be sure to name any theorems you...
For each of the following vector fields, find its curl and
determine if it is a gradient field.
(1 point) For each of the following vector fields, find its curl and determine if it is a gradient field. (a) F = 5(xy + 22) + 10(x2 + y2) 7+ 10(x2 + y2) k. curl F = F ? (b) Ğ = 5yzi + (52z+z2) 7+ (5xy + 2yz) k: curl Ĝ = Ğ ? (c) H = (5xy + yz)...
(1 point) (a) Show that each of the vector fields F-4yi + 4x j, G-i ЗУ x2+y2 x?+yi J, and j are gradient vector fields on some domain (not necessarily the whole plane) x2+y2 by finding a potential function for each. For F, a potential function is f(x, y) - For G, a potential function is g(x, y) - For H, a potential function is h(x, y) (b) Find the line integrals of F, G, H around the curve C...
c. Let F : R³ → R³ be a vector field on R, given by the following function F(x, y, 2) = (x2)i + (y2)J + (xy)k. Calculate the flux of the field across the surface of the hemisphere, : [0, 1] × [0, 2x] → R³, where parametrized by the following function Þ(r, 0) = (r cos 0)i + (r sin 0) + (1 – 1²)!/2 k.
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