For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. 106. F ( x , y ) = 2 x 3 i + 3 y 2 x 2 j
For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. 106. F ( x , y ) = 2 x 3 i + 3 y 2 x 2 j
For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.
106.
F
(
x
,
y
)
=
2
x
3
i
+
3
y
2
x
2
j
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.
For vector field F = (-2y, y), y > 0. Find all points P such that the amount of fluid flowing into P equals the amount of fluid flowing out of P. Write down the equation these points satisfy.
Let A be the vector potential and B the magnetic field of the infinite solenoid of radius R. Then
B(r) = {Bk if R
r
where r = √x² + y² is the distance to the z-axis and B is a constant that depends on the current strength I and the spacing of
the turns of wire.
The vector potential for B is
A(r) = { }B (-3,2,0)
B(-y.x,0)
8
(a) Use Stokes' Theorem to compute the flux of B through a circle in the xy-plane of radius r = 6 R
if r < R
A dr =
(b) Use Stokes' Theorem to compute the circulation of A around the boundary C of a surface lying outside the solenoid.
(Use symbolic notation and fractions where needed.)
C
2
Br
0
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