Zero curl Consider the vector field F = − y x 2 + y 2 i + x x 2 + y 2 j + z k . a. Show that ▿ × F = 0. b. Show that ∮ C F ⋅ d r is not zero on a circle C in the xy -plane enclosing the origin. c. Explain why Stokes’ Theorem does not apply in this case.
Zero curl Consider the vector field F = − y x 2 + y 2 i + x x 2 + y 2 j + z k . a. Show that ▿ × F = 0. b. Show that ∮ C F ⋅ d r is not zero on a circle C in the xy -plane enclosing the origin. c. Explain why Stokes’ Theorem does not apply in this case.
b. Show that
∮
C
F
⋅
d
r
is not zero on a circle C in the xy-plane enclosing the origin.
c. Explain why Stokes’ Theorem does not apply in this case.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Zero curl Consider the vector field
y
F
i +
sj+ zk.
x? + y
x? + y?
a. Show that V × F = 0.
b. Show that fF · dr is not zero on a circle C in the xy-plane
enclosing the origin.
c. Explain why Stokes' Theorem does not apply in this case.
The vector function
r(t)
(5 – 2 sin t) i + (3+ 2 cos t) j + 2 k
-
traces out a circle in 3-space as t varies. In
which plane does this circle lie?
1. plane x
2. plane y
2
3. plane z
= -2
4. plane z
= 2
5. plane x = -2
6. plane y = -2
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