Question

- 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?

Answer #1

(a) Is the vector field F = <e^(−x) cos y, e^(−x) sin y>
conservative?
(b) If so, find the associated potential function φ.
(c) Evaluate Integral C F*dr, where C is the straight line path
from (0, 0) to (2π, 2π).
(d) Write the expression for the line integral as a single
integral without using the fundamental theorem of calculus.

1.) Let f(x,y) =x^2+y^3+sin(x^2+y^3). Determine the line
integral of f(x,y) with respect to arc length over the unit circle
centered at the origin (0, 0).
2.)
Let f ( x,y)=x^3+y+cos( x )+e^(x − y). Determine the line
integral of f(x,y) with respect to arc length over the line segment
from (-1, 0) to (1, -2)

Consider the vector field.
F(x, y, z) =
7ex sin(y), 7ey sin(z), 8ez sin(x)
(a) Find the curl of the vector field.
curl F =
(b) Find the divergence of the vector field.
div F =

Consider the vector field.
F(x, y,
z) =
6ex
sin(y),
7ey
sin(z),
5ez
sin(x)
(a) Find the curl of the vector field.
curl F =
(b) Find the divergence of the vector field.

Consider the vector field. F(x, y, z) = 9ex sin(y), 9ey sin(z),
2ez sin(x) (a) Find the curl of the vector field. curl F = (b) Find
the divergence of the vector field.

Problem 7. Consider the line integral Z C y sin x dx − cos x
dy.
a. Evaluate the line integral, assuming C is the line segment
from (0, 1) to (π, −1).
b. Show that the vector field F = <y sin x, − cos x> is
conservative, and find a potential function V (x, y).
c. Evaluate the line integral where C is any path from (π, −1)
to (0, 1).

Consider the vector force field given by F⃗ = 〈2x + y, 3y +
x〉
(a) Let C1 be the straight line segment from (2, 0) to (−2,
0).
Directly compute ∫ C1 F⃗ · d⃗r (Do not use Green’s Theorem or
the Fundamental Theorem of Line Integration)
(b) Is the vector field F⃗ conservative? If it is not
conservative, explain why. If it is conservative, find its
potential function f(x, y)
Let C2 be the arc of the half-circle...

Consider the vector
field F = ( 2 x e y − 3 ) i + ( x 2 e y + 2 y ) j ,
(a) Find all potential
functions f such that F = ∇ f .
(b) Use (a) to
evaluate ∫ C F ⋅ d r , where C is the curve r ( t ) = 〈 t , t 2 〉 ,
1 ≤ t ≤ 2 .

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.

2. Is the vector field F = < z cos(y), −xz sin(y), x
cos(y)> conservative? Why or why not? If F is conservative, then
find its potential function.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 13 minutes ago

asked 14 minutes ago

asked 20 minutes ago

asked 33 minutes ago

asked 37 minutes ago

asked 52 minutes ago

asked 53 minutes ago

asked 55 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago