Calculus
7th Edition
ISBN: 9781524916817
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 13, Problem 54SP
To determine
To find:Thegiven surface
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1).Evaluate ∮CF•dr, whereF=⟨e^x−y^3,cosy+x^3⟩, and C is a circle of radius 2, centered at the origin, traversed once counterclockwise.
2).Let F(x, y, z) =⟨3x^2y+az, x^3,3x+ 3z^2⟩ be a vector field. For what values of a is F conservative?
8.1
If C is the curve given by r(t)=(1+5sint)i+(1+3sin2t)j+(1+3sin3t)kr(t)=(1+5sint)i+(1+3sin2t)j+(1+3sin3t)k, 0≤t≤π20≤t≤π2 and F is the radial vector field F(x,y,z)=xi+yj+zkF(x,y,z)=xi+yj+zk, compute the work done by F on a particle moving along C.
1) Consider the conservative vector field given by:
F(x, y) = (exy3 + 2e2xy, e2x + 3exy2)
A potential function that generates the vector field F corresponds to:
A) f(x, y) = exy + exy3
B) f(x, y) = 3exy2 +(e2x/2)+(exy4)/4
C) f(x, y) = e2xy + exy3
D) f(x, y) = exy + e2xy3
2) Consider the vector field F(x, y, z) = (y - z sinx, x, 2z + cosx). The work that performs the F field to displace a body, from point A (3π, −1, 1) to point B (π, 2, 0) corresponds approximately to:
A) 28, 45 JB) 32, 42 JC) 15, 71 JD) 13, 72 J
Chapter 13 Solutions
Calculus
Ch. 13.1 - Prob. 1PSCh. 13.1 - Prob. 2PSCh. 13.1 - Prob. 3PSCh. 13.1 - Prob. 4PSCh. 13.1 - Prob. 5PSCh. 13.1 - Prob. 6PSCh. 13.1 - Prob. 7PSCh. 13.1 - Prob. 8PSCh. 13.1 - Prob. 9PSCh. 13.1 - Prob. 10PS
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- Consider I = ∫CF⋅dr, where (img17) is a conservative vector field and curve C is parameterized by:α (t): = ((2 − cos (5t)) cost, (2 − cos (5t)) synt, sin (5t)) with 0≤t≤π. We have that the value of I is equal to: (img18)arrow_forwardConsider the vector field F=(x^2+y^2,10xy). Compute the line integrals ∫c_1 F⋅dr and ∫c_2 F⋅dr, where c_1 (t)=(t, t^2) and c_2 (t)=(t, t) for 0≤t≤1. Can you decide from your answers whether or not F is a gradient vector field? Why or why not?arrow_forward5.Find the circulation of the field F(x, y, z) = (x-y)i+2yj around the circler(t) = costi+ sin tj , 0<=t<=2πarrow_forward
- 1. Let F =i+y^2j +z^4k. Find the general flow lines for this vector field. Present the flow line through the origin.arrow_forward5. Find a potential function corresponding to the vector field F(x, y, z) = (2x, 3Y, 4z).arrow_forwardSuppose initially (t = 0) that the traffic density p = p_0 + epsilon * sinx, where |epsilon| << p_o. Determine p(x, t).arrow_forward
- Find the value of the line integral ∫CF→⋅dr→if F→(x,y)=(y^3+11)i→+(3xy+11)j→is a conservative vector field and Cis the semi circular path from (0,0)to (9,0)shown in the figure below.arrow_forwardThis is a two part problem. Let F = (4xy, 2y^2) be a vector field in the plane, and C the path y = 3x^2 joining (0,0) to (1,3) in the plane. A. Evaluate the line integral F*dr B. Does the integral in part A depend on the path joining (0,0) to (1,3)? Why or why not?arrow_forwardcalc 3 13.7 #5 Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = xy i + yz j + zx kS is the part of the paraboloid z = 2 − x2 − y2 that lies above the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and has upward orientation.arrow_forward
- Let C, V, and£, be the oriented curves in Figure 16, and let F = V f be a gradient vector field such that fc F • dr = 4. What are the values of the following integrals? (a) fv F • dr (b) L F. drarrow_forward(a) Suppose that F is an inverse square force field, that is , F (r) =cr/|r|3 for some constant c, where r=xi+yj+zk . (a) Find the work done by F in moving an object from a pointP1 along a path to a point P2 in terms of the distances d1 and d2 from these points to the origin. (b) An example of an inverse square field is the gravitational field F= -(mMG)r/|r|3 discussed in Example 16.1.4. Use part (a) to find the work done by the gravitational field when the earth moves from aphelion (at a maximum distance of 1.52*108 km from the sun) to perihelion (at a minimum distance of 1.47*108 km). (Use the values m=1.52*1024 kg , M=1.52*1030kg , and G= 6.67*10-11 N.3m2/kg2.) (c) Another example of an inverse square field is the electric force field F=-ɛqQr/|r|3 discussed in Example 16.1.5. Suppose that an electron with a charge of -1.6*10-19 C is located at the origin. A positive unit charge is positioned a distance 10-12 m from the electron and moves to a position half that distance iron: the…arrow_forward6. Consider F = ⟨x, xy, xyz⟩. Does there exist a vector field G such that curl G = F? Why or why not?arrow_forward
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