Homework Answers
6. (4) (a) Is F(x, y, z) = <e'siny, e cosx, esiny > a conservative vector...
6. (4) (a) Is F(x, y, z) = <e'siny, e cosx, esiny > a conservative vector field? Justify. (4) (b) Is F incompressible? Explain. Is it irrotational? Explain. (8) (c) The vector field F(x,y,z)= < 6xy+ e?, 6yx²+zcos(y), sin(y)+xe?> is conservative. Find the potential function f. That is, the function f such that Vf= F. Use a process. Don't guess and check.
(a)If F =< y2 + 2x22, 2xyz, xy² + 2x22 >, find the function f(2, y,...
(a)If F =< y2 + 2x22, 2xyz, xy² + 2x22 >, find the function f(2, y, z) such that F=Vf. (Hint: F is a conservative vector field) (b) A vector field F = Pi +Qj is conservative if Py = Qx. If F = Pi + Qj + Rk, how will you determine if F is conservative? Explain in detail your process and if any underlying conditions are required for your process to work.
Let F(x,y,z) = <2y2z, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative,...
Let F(x,y,z) = <2y2z, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative, find a function f such that F = Vf and f(1,2,1)= 8. (b) Using the result of part(a), evaluate the line integral of F along the following curve C from (0, 0, 0) to (3.9, 1.4, 2.6). y2 + x4z3 + 2xy(x3 + y4 + 24)1/3 = K ; K is a constant Answer: Next page
7. Is F(x, y) =< 2x, 6y? > a conservative vector field?
7. Is F(x, y) =< 2x, 6y? > a conservative vector field?
96. Consider a vector field F(x, y, z) =< x + x cos(yz), 2y - eyz,...
96. Consider a vector field F(x, y, z) =< x + x cos(yz), 2y - eyz, z- xy > and scalar function f(x, y, z) = xy3e2z. Find the following, or explain why it is impossible: a) gradF (also denoted VF) b) divF (also denoted .F) c) curl(f) (also denoted xf) d) curl(gradf) (also denoted V x (0f) e) div(curlF) (also denoted 7. (V x F))
Vector Fields problems. Question 6 Let f (x, y) = x3 + 4xy + 2y2 and...
Vector Fields problems.
Question 6 Let f (x, y) = x3 + 4xy + 2y2 and of =<P,Q >. P (2, 1) = ? Question 7 vf same as in Question 6 Q (2, 1) =?
Is the below statement True or False? The vector field F(x,y) =<xy?, x?y) is conservative. True...
Is the below statement True or False? The vector field F(x,y) =<xy?, x?y) is conservative. True False
3) Given vector field F(x,y,z)=<y, xz,x? >. Find N dr where T is the path around...
3) Given vector field F(x,y,z)=<y, xz,x? >. Find N dr where T is the path around the triangle with vertices (1,0,0),(0,1,0) and (0,0,1) traced counterclockwise (when viewed from above.)
8Two vector fields are given: F(x,y,z) - (esin(yz), ze* cos(yz), ye* cos(yz)) and F(x,y,z) = (z...
8Two vector fields are given: F(x,y,z) - (esin(yz), ze* cos(yz), ye* cos(yz)) and F(x,y,z) = (z cos y, xz sin y, x cos y). a) Determine which vector field above is conservative. Justify. Foly = fjol so, <ea sin(J2), 20% cos(82), y acos (92)) Conservative. b) For the vector field that is conservative, find a function f such that F - Vf. Lxelsing2, zetos yea, yet cosy 2 c) Use the Fundamental Theorem of Line Integrals to find the work...
D Question 11 12 pts to Consider the vector field F (x, y, z) =< 2x...
D Question 11 12 pts to Consider the vector field F (x, y, z) =< 2x – yz, 2y – az,2z – xy>. a) (3) Is this vector field conservative? Justify your answer. b) (9) Find the amount of work done by this vector field in moving a particle along the curve (t) =< 3cost, cos’t, cos” (2t) > from t = 0 tot = 1
6. (4) (a) Is F(x, y, z) = <e'siny, e cosx, esiny > a conservative vector field? Justify. (4) (b) Is F incompressible? Explain. Is it irrotational? Explain. (8) (c) The vector field F(x,y,z)= < 6xy+ e?, 6yx²+zcos(y), sin(y)+xe?> is conservative. Find the potential function f. That is, the function f such that Vf= F. Use a process. Don't guess and check.
(a)If F =< y2 + 2x22, 2xyz, xy² + 2x22 >, find the function f(2, y, z) such that F=Vf. (Hint: F is a conservative vector field) (b) A vector field F = Pi +Qj is conservative if Py = Qx. If F = Pi + Qj + Rk, how will you determine if F is conservative? Explain in detail your process and if any underlying conditions are required for your process to work.
Let F(x,y,z) = <2y2z, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative, find a function f such that F = Vf and f(1,2,1)= 8. (b) Using the result of part(a), evaluate the line integral of F along the following curve C from (0, 0, 0) to (3.9, 1.4, 2.6). y2 + x4z3 + 2xy(x3 + y4 + 24)1/3 = K ; K is a constant Answer: Next page
7. Is F(x, y) =< 2x, 6y? > a conservative vector field?
96. Consider a vector field F(x, y, z) =< x + x cos(yz), 2y - eyz, z- xy > and scalar function f(x, y, z) = xy3e2z. Find the following, or explain why it is impossible: a) gradF (also denoted VF) b) divF (also denoted .F) c) curl(f) (also denoted xf) d) curl(gradf) (also denoted V x (0f) e) div(curlF) (also denoted 7. (V x F))
Vector Fields problems.
Question 6 Let f (x, y) = x3 + 4xy + 2y2 and of =<P,Q >. P (2, 1) = ? Question 7 vf same as in Question 6 Q (2, 1) =?
Is the below statement True or False? The vector field F(x,y) =<xy?, x?y) is conservative. True False
3) Given vector field F(x,y,z)=<y, xz,x? >. Find N dr where T is the path around the triangle with vertices (1,0,0),(0,1,0) and (0,0,1) traced counterclockwise (when viewed from above.)
8Two vector fields are given: F(x,y,z) - (esin(yz), ze* cos(yz), ye* cos(yz)) and F(x,y,z) = (z cos y, xz sin y, x cos y). a) Determine which vector field above is conservative. Justify. Foly = fjol so, <ea sin(J2), 20% cos(82), y acos (92)) Conservative. b) For the vector field that is conservative, find a function f such that F - Vf. Lxelsing2, zetos yea, yet cosy 2 c) Use the Fundamental Theorem of Line Integrals to find the work...
D Question 11 12 pts to Consider the vector field F (x, y, z) =< 2x – yz, 2y – az,2z – xy>. a) (3) Is this vector field conservative? Justify your answer. b) (9) Find the amount of work done by this vector field in moving a particle along the curve (t) =< 3cost, cos’t, cos” (2t) > from t = 0 tot = 1
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