Calculus
6th Edition
ISBN: 9781465208880
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 13.3, Problem 35PS
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1. Find the line integral in a vector field
F. dr, where F = (y, x + 2y) and C is
a curve consisting of 2 parts. First, the straight line from (-1,1) to (1,1), followed by
the parabola y = x² from (1,1) to (2,4). You must show all your work.
5. Show that F= (y cos(x),sin(x)+ y) is a conservative vector field and use this to
calculate the work done by F in moving a particle from (7/4,1) to (27/3,3).
2. Let r = √x² + y² + z².
(a) For n ≥ 1 and r > 0, express V(r-n) in terms of r and the radial vector field er.
(b) For r> 0, express Vlnr in terms of r and the radial vector field er.
(c) Taking A as in question 1, show that for r> 0 we have A(-¹) = 0.
Chapter 13 Solutions
Calculus
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- Suppose we have a vector field Ē(x, y, z) defined by E = -Vộ. Prove or disprove the following assertion: V(V · Ē) = v²Ẽ (4) It might help to write out the components of E in terms of derivatives of ø.arrow_forward19. (a) Evaluate the line integral ſ F · dr, for F(x, y) = ex−¹ i + xy j and C is given by r(t) = t² i + t³ j, 0 ≤ t ≤ 1. (b) Illustrate part (a) by using a graphing calculator or computer to graph C and the vectors from the vector field corresponding to t = 0,1/2 and 1 (as in Figure 13).arrow_forwardMatch each of the functions (a)-(c) with the corresponding gradient vector field from those in (I)-(VI). Enter your answer as upper case roman numerals: I, II, III, etc. (a) f(x, y) = x cos(y) (b) f(x, y) = (x - 2)²(y-2)/5 (c) f(x, y) = sin(x) · cos(y) (click on an image to enlarge it) IV II V II VIarrow_forward
- 13. Consider the vector field, F(x, y, z) = 1, 22 + y²' z² + y²,arrow_forward2. Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such that [₁ C₁ F. dr = 0 and [₁ C₂ F. dr = 1arrow_forward3. Show that the vector field F = -C with C> 0, is conservative and find its potential.arrow_forward
- 1. Let F(x, y, z) be the vector field (x²(1 + y²z²), Q(x, y, z), ey cos(22)), where is a differentiable function of x, y, and z. Suppose div F = 2x(1 + y2² 2²) — 2ery sin(22) + z tan(2y+z) + 2z sec² (2y + 2); and - • curl F = (xey cos(22)-y tan(2y+z)-y sec² (2y+z), 2x²y²z-yery cos(22), -2x²yz²). Determine div(Q).arrow_forward7. Sketch the vector field F = (x − y)î + yĵ. Identify where F₁ vanishes. Identify where F2 vanishes. Plot a few flow lines. 8. Repeat the previous problem for F = yî + (x − y)î.arrow_forward7. Find the outward flux of the field F(x, y) = -yi + x² j across the closed counter-clockwise curve consisting of the top semi-circle of + y? = 4 followed by the straight line from (-2,0) to (2,0).arrow_forward
- please help mearrow_forward1. Let f(x, y) cos(-y) and C be the line segment from (-1, 1) to (3,-2). Set up but do NOT evaluate the following. 2. = √ 1(2₁0 f(x,y) ds Note to your future self about how to do this problem: Consider the vector field F(x, y) = (x,y). Let C be the upper half of the circle of radius 1 centered at the origin (² + y² = 1) that is traced once counterclockwise. Evaluate the line integral: SF. di Note to your future self about how to do this problem: (Jaarrow_forward5. Let F be the gradient vector field of the function f(x, y) = 4.xy + 3y?. Evaluate SF. dr, where C still has endpoints (2,0) and (0, 2) as in the last problem, but is moving counterclockwise along a quarter of the circle of radius 2 centered at the origin.arrow_forward
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