Question

Suppose you are looking at a field [m[x,y],n[x,y]] which has one singualrity at a point P and no other singularities. While studying the singulairty, you center a circle of radius r on the point P and no other singularites. While studying the singularity, you center a circle of radius r on the point P and parameterize this circle in the counter-clockwise direction as Cr;{x[t],y[t]}. You calculate (integralCr) -n[x[t],y[t]]x'[t] + m[x[t], y[t]]y'[t]dt and find that it is equal to -1 + e^(r^2)

What is this integral telling you about the vector field at the point P

Answer #1

Consider the vector field F = <2 x
y^3 , 3 x^2
y^2+sin y>. Compute
the line integral of this vector field along the quarter-circle,
center at the origin, above the x axis, going from the point (1 ,
0) to the point (0 , 1). HINT: Is there a potential?

A solenoid with 7 x 104 /m (turns per meter) runs through the
x-y plane. Within the solenoid a charge of +3 coulombs with a mass
of 4 grams is initially moving at 39 m/s in the +x direction. Due
to the magnetic field created by the solenoid it orbits in a circle
of radius 0.34 meters counter-clockwise. What's the amount, in
amps, and direction of the current for the solenoid? If clockwise,
input the current as negative.

1.Let y=6x^2. Find a parametrization of the
osculating circle at the point x=4.
2. Find the vector OQ−→− to the center of the
osculating circle, and its radius R at the point
indicated. r⃗
(t)=<2t−sin(t),
1−cos(t)>,t=π
3. Find the unit normal vector N⃗ (t)
of r⃗ (t)=<10t^2, 2t^3>
at t=1.
4. Find the normal vector to r⃗
(t)=<3⋅t,3⋅cos(t)> at
t=π4.
5. Evaluate the curvature of r⃗
(t)=<3−12t, e^(2t−24),
24t−t2> at the point t=12.
6. Calculate the curvature function for r⃗...

Consider the vector field below: F ⃗=〈2xy+y^2,x^2+2xy〉 Let C be
the circular arc of radius 1 starting at (1,0), oriented counter
clock wise, and ending at another point on the circle. Determine
the ending point so that the work done by F ⃗ in moving an object
along C is 1/2.

Suppose that you have a circular magnetic field ‘B’ of radius R
= 9.0 cm, pointing inside the page and if this field is increased
at the rate dB/dt = 0.15 T/sec. calculate (a) the magnitude of
induced electric field at a point within the magnetic field at a
distance r =5.0 cm, from the center of the field and (b) the
magnitude of the induced electric field at a point out side the
magnetic field at a distance r...

For each vector field F~ (x, y) = hP(x, y), Q(x, y)i, find a
function f(x, y) such that F~ (x, y) = ∇f(x, y) = h ∂f ∂x , ∂f ∂y i
by integrating P and Q with respect to the appropriate variables
and combining answers. Then use that potential function to directly
calculate the given line integral (via the Fundamental Theorem of
Line Integrals):
a) F~ 1(x, y) = hx 2 , y2 i Z C F~ 1...

Define p to be the set of all pairs (l,m) in N×N such that l≤m.
Which of the conditions (a), (c), (r), (s), (t) does p satisfy?
(a) For any two elements y and z in X with (y,z)∈r and (z,y)∈r,
we have y=z
.(c) For any two elements y and z in X, we have (y,z)∈r or
(z,y)∈r.
(r) For each element x in X, we have (x,x)∈r.
(s) For any two elements y and z in X with...

GOAL Use the superposition principle to calculate the electric
field due to two point charges. Consider the following
figure.
The resultant electric field at P equals the
vector sum 1 + 2, where 1 is the
field due to the positive charge q1and
2 is the field due to the negative charge
q2.Two point charges lie along the
x-axis in the x y-coordinate plane.
Positive charge q1 is at the origin, and
negative charge q2 is at (0.300 m, 0). Point...

(1 point) A Bernoulli differential equation is one of the
form
dydx+P(x)y=Q(x)yn (∗)
Observe that, if n=0 or 1, the Bernoulli equation is linear. For
other values of n, the substitution u=y1−n transforms the Bernoulli
equation into the linear equation
dudx+(1−n)P(x)u=(1−n)Q(x).dudx+(1−n)P(x)u=(1−n)Q(x).
Consider the initial value problem
y′=−y(1+9xy3), y(0)=−3.
(a) This differential equation can be written in the form (∗)
with
P(x)= ,
Q(x)= , and
n=.
(b) The substitution u= will transform it into the linear
equation
dudx+ u= .
(c) Using...

(1 point) Given the following differential equation
(x2+2y2)dxdy=1xy,
(a) The coefficient functions are M(x,y)= and N(x,y)= (Please input
values for both boxes.)
(b) The separable equation, using a substitution of y=ux, is
dx+ du=0 (Separate the variables with x with dx only and u with du
only.) (Please input values for both boxes.)
(c) The solution, given that y(1)=3, is
x=
Note: You can earn partial credit on this
problem.
I just need part C. thank you

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