Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)
6th Edition
ISBN: 9781524908102
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 13.3, Problem 33PS
To determine
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2. 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 = 1
13. Let
V = xi+yj
(a) Give reasons why V(x, y) is a vector field
(b) Sketch the vector field V(x, y)
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.
Chapter 13 Solutions
Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)
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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- 5. Consider the vector field F(x,y,z) = yzi +xzj+(xy+2z) k and the curve C given by r(t) t + sin at t + cos -nt 2t 0arrow_forward2. 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_forward9. The outward flux of the vector field F=(x,0, z) across the cylinder x + y² = 1, for 0arrow_forward6. Let F = (2.xy)i + (x²)j be the force field (a) Show that F is conservative. A(-1, 2) В 3, 1) 1 3arrow_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_forward7.3 Compute the curl of the vector field F⃗ =⟨x4,y4,z5⟩F→=⟨x4,y4,z5⟩.curl(F⃗ (x,y,z))curl(F→(x,y,z)) = What is the curl at the point (−4,3,5)(−4,3,5)?curl(F⃗ (−4,3,5))curl(F→(−4,3,5)) = Is this vector field irrotational (curl free) or not?arrow_forwardarrow_back_iosarrow_forward_iosRecommended textbooks for you
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