Question

Let F ( x , y ) = 〈 e^x + y^2 − 3 , − e ^(− y) + 2 x y + 4 y 〉. a) Determine if F ( x , y ) is a conservative vector field and, if so, find a potential function for it. b) Calculate ∫ C F ⋅ d r where C is the curve parameterized by r ( t ) = 〈 2 t , 4 t + sin π t 〉 for 0 ≤ t ≤ 3

Answer #1

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1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

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 .

(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.

a. Is F(x,y,z)= <(e^z)siny,(e^z)cosx,(e^x)siny> a
conservative vector field? Justify.
b. Is F incompressible? Explain. Is it irrotational?
Explain.
c. The vector field F(x,y,z)= < 6xy^2+e^z, 6yx^2
+zcos(y),sin(y)xe^z > is conservative. Find the potential
function f. That is, the function f such that ▽f=F. Use a
process.

The function f(x, y) = x^−2 y^3 is a potential for a vector
field F. Use this to evaluate ∫ C F · dr where C is a curve from
(1, 1) to (2, 2).

B.) Let R be the region between the curves y = x^3 , y = 0, x =
1, x = 2. Use the method of cylindrical shells to compute the
volume of the solid obtained by rotating R about the y-axis.
C.) The curve x(t) = sin (π t) y(t) = t^2 − t has two tangent
lines at the point (0, 0). List both of them. Give your answer in
the form y = mx + b ?...

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.

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).

Let y = x 2 + 3 be a curve in the plane.
(a) Give a vector-valued function ~r(t) for the curve y = x 2 +
3.
(b) Find the curvature (κ) of ~r(t) at the point (0, 3). [Hint:
do not try to find the entire function for κ and then plug in t =
0. Instead, find |~v(0)| and dT~ dt (0) so that κ(0) = 1 |~v(0)|
dT~ dt (0) .]
(c) Find the center and...

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?

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