
Developmental Mathematics (9th Edition)
9th Edition
ISBN: 9780321997173
Author: Marvin L. Bittinger, Judith A. Beecher
Publisher: PEARSON
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Chapter I, Problem 3ES
To determine
To calculate: The distance between points
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(3) (16 points) Consider
z = uv,
u = x+y,
v=x-y.
(a) (4 points) Express z in the form z = fog where g: R² R² and f: R² →
R.
(b) (4 points) Use the chain rule to calculate Vz = (2, 2). Show all intermediate
steps otherwise no credit.
(c) (4 points) Let S be the surface parametrized by
T(x, y) = (x, y, ƒ (g(x, y))
(x, y) = R².
Give a parametric description of the tangent plane to S at the point p = T(x, y).
(d) (4 points) Calculate the second Taylor polynomial Q(x, y) (i.e. the quadratic
approximation) of F = (fog) at a point (a, b). Verify that
Q(x,y) F(a+x,b+y).
=
(6) (8 points) Change the order of integration and evaluate
(z +4ry)drdy .
So S√ ²
0
(10) (16 points) Let R>0. Consider the truncated sphere S given as
x² + y² + (z = √15R)² = R², z ≥0.
where F(x, y, z) = −yi + xj .
(a) (8 points) Consider the vector field
V (x, y, z) = (▼ × F)(x, y, z)
Think of S as a hot-air balloon where the vector field V is the velocity vector
field measuring the hot gasses escaping through the porous surface S. The flux
of V across S gives the volume flow rate of the gasses through S. Calculate
this flux.
Hint: Parametrize the boundary OS. Then use Stokes' Theorem.
(b) (8 points) Calculate the surface area of the balloon. To calculate the surface
area, do the following:
Translate the balloon surface S by the vector (-15)k. The translated
surface, call it S+ is part of the sphere x² + y²+z² = R².
Why do S and S+ have the same area?
⚫ Calculate the area of S+. What is the natural spherical parametrization
of S+?
Chapter I Solutions
Developmental Mathematics (9th Edition)
Ch. I - Find the distance between each pair of points....Ch. I - Prob. 2DECh. I - Prob. 3DECh. I - Prob. 4DECh. I - Prob. 1ESCh. I - Prob. 2ESCh. I - Prob. 3ESCh. I - Prob. 4ESCh. I - Prob. 5ESCh. I - Prob. 6ES
Ch. I - Prob. 7ESCh. I - Prob. 8ESCh. I - Prob. 9ESCh. I - Prob. 10ESCh. I - Prob. 11ESCh. I - Prob. 12ESCh. I - Prob. 13ESCh. I - Prob. 14ESCh. I - Prob. 15ESCh. I - Prob. 16ESCh. I - Prob. 17ESCh. I - Prob. 18ESCh. I - Prob. 19ESCh. I - Prob. 20ESCh. I - Prob. 21ESCh. I - Prob. 22ESCh. I - Prob. 23ESCh. I - Prob. 24ESCh. I - Prob. 25ESCh. I - Prob. 26ESCh. I - Prob. 27ESCh. I - Prob. 28ESCh. I - Prob. 29ESCh. I - Prob. 30ESCh. I - Prob. 31ESCh. I - Prob. 32ESCh. I - Prob. 33ESCh. I - Prob. 34ESCh. I - Prob. 35ESCh. I - Prob. 36ESCh. I - Prob. 37ESCh. I - Prob. 38ESCh. I - Prob. 39ESCh. I - Prob. 40ESCh. I - Prob. 41ESCh. I - Prob. 42ESCh. I - Prob. 43ESCh. I - Prob. 44ES
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