1. is a vector field F(a,v) which has the property curl(F) div(F)-C, where C is
ID: 2879313 • Letter: 1
Question
1. is a vector field F(a,v) which has the property curl(F) div(F)-C, where C is a some constant. 2. Assume a vector field F(z,v) is a gradient field, then JC Fds 0 if C is a circle with center (0,0, 1), radius 1, located on z 1. 3. If the flux of vector field is zero at any point of surface S in space, then the divergence of the field is zero everywhere in space. 4. The line integral of the curl of a vector field F(r,v) along a circle in the zw- plane is 5. If a vector field F(z,v) atisfies curl(F)-0 for all z, then F(z,v) is a conservative field. 6. A vector field F(z, y) is constant if and only if the divergence and the curl of it are both zero. 7. The divergence of a conservative vector field is always zero. 8. The line integral of the vector feld F(z, y) z, y, z along a line segment from (0,0,0) to (1,1,1) is 1 9. Function f(z, y, z) ryz is potential for yz,zz,zy continuous functions. 11. The scalar line integral doesn't depend on how the curve is parametrized. 12. If the orientation of the curve is changed, the vector line integral changes its sign. 13. If the orientation of the curve is changed, the scalar line integral changes its sign. 15. The line integral fordy yde equals the area enclosed by C. 16. For any curve C, fo Fdz Gdy- Fv)dA. 17. Green's formula relates the circulation around the boundary to the surface integral of the curl. 18. The flux of curl(F) through all surfaces is zero 19. div (curl(F)) (div (F)) 0 for any vector field F(z,y, z). curl 20. The flux of F(r, y, z) y, is positive through any closed surface.Explanation / Answer
12)
TRUE because the vector is dependent on direction.
Reverse the direction and we get a negative answer
13)
FALSE because a scalar field will not have be affected
by the direction
15)
FALSe. The actual formula will include a 1/2 multiplied
to the integral
16)
TRUE because this is green's theorem statement
17)
FALSE. That would be the Stoke's theorem
18)
FALSe certainly.
19)
TRUE
20) TRUE
Since we have a closed surface here , we can use divergence theorem
and write F.dS =triple integral (div F . dV)
The triple integral will be positive if div F is positive
We wil find div F = 3x^2 + 3y^2 + 3z^2, which is ALWAYS positive
So, div F . dV will always be positive and thus flux always positive here
8) FALSE
WE can parameterize this line as <t,t,t> with 0 <= t <= 1
r' = <1,1,1>dt
So, now, F . dr becomes :
<x,y,z> . <1,1,1>
x+y+z
Plug in for x,y,z with t, we get :
t+t+t
3t
So, integral of 3t from t = 0to 1
3t^2 (from 0 to 1)
3(1^2 - 0^2)
3
So, the line integral aint 1, it is 3
7) FALSE.
Only the curl of a conservative field is always 0
10) FALSE
A potential function exists only if curl F = 0, i.e only if the field F is conservative
11) FALSE
Depends very much on the path over which it is parameterized
Only a conservative field will not depend on the parameterization because a conservative field is path independent
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.