(a) div F (b) curl F 2. Let F(r, y, z) = (x, y, z) and let g(x, y, z) |F(x, y, z)|. Find values of a, b, c, and m that satisfy the following: %3D (a) div F = a (b) V (gF) =bg (c) |curl (gF)| = c (d) V² (9*) = m g Note: Vf = div(Vf) = V · (Vf) = faz + Syy + fzz is the Laplacian of f. 3. Let F(r, y, z) = P(x, y, z)i+ Q(x,y, z)j+ R(x, y, z)k. The vector Laplacian oper- ator is defined as V°F = V (V · F) – V × (V × F). This is a rather involved definition that can be simplified. By expanding the right-hand side of this equation, show that an equivalent definition of the vector Laplacian operator is V°F = (V°P) i+ (v²Q)j+(V°R)k. 4. Find the surface area of the portion of the paraboloid r = 4 - y - 2 that lies between the cylinders y?+ 22 = 1 and y? + 22 = 4. Over
(a) div F (b) curl F 2. Let F(r, y, z) = (x, y, z) and let g(x, y, z) |F(x, y, z)|. Find values of a, b, c, and m that satisfy the following: %3D (a) div F = a (b) V (gF) =bg (c) |curl (gF)| = c (d) V² (9*) = m g Note: Vf = div(Vf) = V · (Vf) = faz + Syy + fzz is the Laplacian of f. 3. Let F(r, y, z) = P(x, y, z)i+ Q(x,y, z)j+ R(x, y, z)k. The vector Laplacian oper- ator is defined as V°F = V (V · F) – V × (V × F). This is a rather involved definition that can be simplified. By expanding the right-hand side of this equation, show that an equivalent definition of the vector Laplacian operator is V°F = (V°P) i+ (v²Q)j+(V°R)k. 4. Find the surface area of the portion of the paraboloid r = 4 - y - 2 that lies between the cylinders y?+ 22 = 1 and y? + 22 = 4. Over
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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How do you do #3
Transcribed Image Text:1. Given the vector field F =
evaluate:
(a) div F
(b) curl F
2. Let F(x, y, z) = (x, y, z) and let g(x, y, z) = |F(x, y, z)|. Find values of a, b, c, and
m that satisfy the following:
(a) div F = a
(b) V (gF) = bg
(c) | curl (gF)| = c
(d) V? (9³) =
m g
Note: Vf = div(Vf) = V · (Vf) = faz + Syy + fzz is the Laplacian of f.
3. Let F(x, y, z) = P(x, y, z)i+ Q(x,y, z)j+ R(x, y, z)k. The vector Laplacian oper-
ator is defined as
V'F = V (V · F) - V × (V × F).
This is a rather involved definition that can be simplified. By expanding the
right-hand side of this equation, show that an equivalent definition of the vector
Laplacian operator is
V°F = (V°P) i+ (V²Q)j+(V°R) k.
4. Find the surface area of the portion of the paraboloid a = 4 - y² – 2² that lies
between the cylinders y? + z2 = 1 and y? + z2 = 4.
Over
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