- #1
LCSphysicist
- 644
- 162
- Homework Statement
- ...
- Relevant Equations
- ...
##F = (P,Q,R)## is a field of vector C1 defined on ##V = R3-{0,0,0}##
There are a lot of true or false statement here. I am a little skeptical about my answer because it contains a lot of F, but let's go.
1 Rot of F is null in V iff ##\int \int_{S} P dx + Q dy + R dz = 0## for all sphere S belonging to V oriented outside.
2 F is conservative in V iff rot of F is null in V
3 P,Q and R are positive iff ##\int \int_{S} P dx + Q dy + R dz > 0## for all sphere S belonging to V oriented outside.
4 F is gradient in V iff ##\int \int_{S} P dx + Q dy + R dz = 0## for all sphere S belonging to V oriented outside.
5 F is null in V iff ##\int \int_{S} P dx + Q dy + R dz = 0## for all sphere S belonging to Voriented outside.
6 rot of F is null in V iff ##\int \int_{C} P dx + Q dy + R dz = 0# for all circumference oriented with C belonging to V
7 P,Q,R are positie iff ##\int \int_{C} P dx + Q dy + R dz>0## for all circumference C belonging to V
8 F is gradient in V iff ##\int \int_{C} P dx + Q dy + R dz = 0## for all circumference C belonging to V
9 F is null in V iff ##\int \int_{C} P dx + Q dy + R dz = 0## for all circumference C belonging to V
To be pretty honest, i am not sure if the author really want to means "if" instead of "iff", or if he commited a typo writing "circumference C" instead of "closed curve C". But in the way it is, i found too much F! see
1 F
2 T
(3 ... 9) F
So only the 2 seems true to me, since R3 minus origin is simply connected and the vector is C1.
All the others answer would be really different if C were just a closed curve, not a circumference. Or if S were a closed surface, not a sphere.
I have some doubts, are my answers right? Or there is some special proeprty about circumference that i am missing that changes all the answer.
There are a lot of true or false statement here. I am a little skeptical about my answer because it contains a lot of F, but let's go.
1 Rot of F is null in V iff ##\int \int_{S} P dx + Q dy + R dz = 0## for all sphere S belonging to V oriented outside.
2 F is conservative in V iff rot of F is null in V
3 P,Q and R are positive iff ##\int \int_{S} P dx + Q dy + R dz > 0## for all sphere S belonging to V oriented outside.
4 F is gradient in V iff ##\int \int_{S} P dx + Q dy + R dz = 0## for all sphere S belonging to V oriented outside.
5 F is null in V iff ##\int \int_{S} P dx + Q dy + R dz = 0## for all sphere S belonging to Voriented outside.
6 rot of F is null in V iff ##\int \int_{C} P dx + Q dy + R dz = 0# for all circumference oriented with C belonging to V
7 P,Q,R are positie iff ##\int \int_{C} P dx + Q dy + R dz>0## for all circumference C belonging to V
8 F is gradient in V iff ##\int \int_{C} P dx + Q dy + R dz = 0## for all circumference C belonging to V
9 F is null in V iff ##\int \int_{C} P dx + Q dy + R dz = 0## for all circumference C belonging to V
To be pretty honest, i am not sure if the author really want to means "if" instead of "iff", or if he commited a typo writing "circumference C" instead of "closed curve C". But in the way it is, i found too much F! see
1 F
2 T
(3 ... 9) F
So only the 2 seems true to me, since R3 minus origin is simply connected and the vector is C1.
All the others answer would be really different if C were just a closed curve, not a circumference. Or if S were a closed surface, not a sphere.
I have some doubts, are my answers right? Or there is some special proeprty about circumference that i am missing that changes all the answer.
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