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 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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- 1. An illustration of the vector field F(r, y) = aj is given, (a) Find paths C1, C2, and C3 from P to Q such that | F. = 0, F. > 0, [ F.a < 0 (b) Is F a gradient field? Explain.arrow_forward9. Sketch the vector field F(x, y) 1 -(yi — хі). Vx? + y?arrow_forward13. Consider the vector field, F(x, y, z) = 1, 22 + y²' z² + y²,arrow_forward
- 2. Let f : R" → R be a continuously differentiable function, while a E R" is its non-stationary point (i.e., Vf(x) # 0). Moreover, let d* be the vector defined by Vf(x) ||Vf(x)||' d* = - while d is any unit vector in R" different from d* (i.e., d e R", ||d|| = 1, d + d*). Show that there exist real numbers d E (0, 00), ɛ E (0, 0) such that %3D f(x + td*) – f(x + td) 0. (ii) For t z 0, approximate f(x+td*) and f(x+td) using the first-order Taylor polynomial of f(x) at x.arrow_forwardDetermine the signs of the line integrals for the pictured vector fields and curves. C₂ (a) (b) C₁ (c) √67 F.dr dr ---Select--- 6.7. 7. dr---Select--- V C₂ √₂7. d² dr (d) Sc.7. dr -- Select.. ---Select--- ✓ Oarrow_forward7. A vector field is specified as F= 2(x + y) sin лza, - (x² + y)a, + [10/(x² + y²)]a.. Specify the locus of all points at which: (a) F, = 0; (b) F, = 0; (c) |F₁ = 1.arrow_forward
- 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).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_forwardTrue or false 1. If F =< P(x,y),Q(x,y) > and Py = Qx in an open region D, then F is conservative. 2. If F and G are vector fields, then curl(F ·G) = curl F ·curl G. 3. It is impossible to find three non-zero three-dimensional real vectors u, v, and w such that(v ·w)(v ·u) = 0 while simultaneously (v ×w) ·(v ×u) = 0. 4. Recall that κ(t) denotes the curvature of a curve. If κ(t) = d for all t, with d ≥ 0 , then thecurve is a straight line.arrow_forward
- 4. The vector field (3r²y + y +2, x³ + x − 1) is conservative. Find a function f such that Vƒ = F.arrow_forward3. Show that F(x, y, z) = (y cos (x))i + (x sin (y))j + (xyz)k is not a gradient field. 4. Show that V(x, y, z) = (xsin y, y cos z, xyz) can't be written as the curl of another vector field F. → 5. Compute Af for f: R³ R defined by f(x, y, z) = x sin z+y cos z+e. Is this function harmonic? Why or why not?arrow_forward
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