Bundle: Differential Equations with Boundary-Value Problems, Loose-leaf Version, 9th + WebAssign Printed Access Card for Zill's Differential Equations ... Problems, 9th Edition, Single-Term
9th Edition
ISBN: 9781337604901
Author: Dennis G. Zill
Publisher: Cengage Learning
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Chapter 11.2, Problem 24E
To determine
The complex form of the Fourier series of given function.
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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 11 Solutions
Bundle: Differential Equations with Boundary-Value Problems, Loose-leaf Version, 9th + WebAssign Printed Access Card for Zill's Differential Equations ... Problems, 9th Edition, Single-Term
Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - Prob. 6ECh. 11.1 - Prob. 7ECh. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...
Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 13 and 14 verify by direct integration...Ch. 11.1 - In Problems 13 and 14 verify by direct integration...Ch. 11.1 - Let {n(x)} be an orthogonal set of functions on...Ch. 11.1 - Let {n(x)} be an orthogonal set of functions on...Ch. 11.1 - Let {n(x)} be an orthogonal set of functions on...Ch. 11.1 - From Problem 1 we know that f1(x) = x and f2(x) =...Ch. 11.1 - A real-valued function is said to be periodic with...Ch. 11.1 - A real-valued function is said to be periodic with...Ch. 11.1 - Prob. 21ECh. 11.1 - A real-valued function is said to be periodic with...Ch. 11.1 - A real-valued function is said to be periodic with...Ch. 11.1 - A real-valued function is said to be periodic with...Ch. 11.1 - Prob. 25ECh. 11.1 - Prob. 26ECh. 11.1 - Prob. 27ECh. 11.1 - Relate the orthogonal set B in Problem 27 with a...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 1–16 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - Prob. 13ECh. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 116 find the Fourier series of f on...Ch. 11.2 - In Problems 17 and 18 sketch the periodic...Ch. 11.2 - In Problems 17 and 18 sketch the periodic...Ch. 11.2 - Use the result of Problem 5 to show that...Ch. 11.2 - Prob. 20ECh. 11.2 - Use the result of Problem 7 to show that...Ch. 11.2 - Prob. 22ECh. 11.2 - Prob. 23ECh. 11.2 - Prob. 24ECh. 11.3 - In Problems 110 determine whether the function is...Ch. 11.3 - Prob. 2ECh. 11.3 - In Problems 110 determine whether the function is...Ch. 11.3 - In Problems 110 determine whether the function is...Ch. 11.3 - Prob. 5ECh. 11.3 - Prob. 6ECh. 11.3 - Prob. 7ECh. 11.3 - Prob. 8ECh. 11.3 - Prob. 9ECh. 11.3 - Prob. 10ECh. 11.3 - In Problems 1124 expand the given function in an...Ch. 11.3 - In Problems 1124 expand the given function in an...Ch. 11.3 - In Problems 1124 expand the given function in an...Ch. 11.3 - In Problems 1124 expand the given function in an...Ch. 11.3 - Prob. 15ECh. 11.3 - In Problems 11-24 expand the given function in an...Ch. 11.3 - Prob. 17ECh. 11.3 - In Problems 11-24 expand the given function in an...Ch. 11.3 - In Problems 11-24 expand the given function in an...Ch. 11.3 - In Problems 1-10 determine whether the function is...Ch. 11.3 - In Problems 1-10 determine whether the function is...Ch. 11.3 - In Problems 1124 expand the given function in an...Ch. 11.3 - In Problems 1124 expand the given function in an...Ch. 11.3 - Prob. 24ECh. 11.3 - In Problems 2534 find the half-range cosine and...Ch. 11.3 - Prob. 26ECh. 11.3 - Prob. 27ECh. 11.3 - In Problems 2534 find the half-range cosine and...Ch. 11.3 - Prob. 29ECh. 11.3 - Prob. 30ECh. 11.3 - Prob. 31ECh. 11.3 - In Problems 2534 find the half-range cosine and...Ch. 11.3 - In Problems 2534 find the half-range cosine and...Ch. 11.3 - Prob. 34ECh. 11.3 - In Problems 3538 expand the given function in a...Ch. 11.3 - In Problems 3538 expand the given function in a...Ch. 11.3 - Prob. 37ECh. 11.3 - In Problems 3538 expand the given function in a...Ch. 11.3 - Prob. 39ECh. 11.3 - In Problems 3942 suppose the function y = f(x), 0 ...Ch. 11.3 - In Problems 3942 suppose the function y = f(x), 0 ...Ch. 11.3 - Prob. 42ECh. 11.3 - In Problems 43 and 44 proceed as in Example 4 to...Ch. 11.3 - In Problems 43 and 44 proceed as in Example 4 to...Ch. 11.3 - Prob. 45ECh. 11.3 - Prob. 46ECh. 11.3 - Suppose a uniform beam of length L is simply...Ch. 11.3 - Prob. 50ECh. 11.3 - Prob. 51ECh. 11.3 - Prob. 52ECh. 11.3 - Prob. 53ECh. 11.3 - Prob. 54ECh. 11.4 - Consider y + y = 0 subject to y(0) = 0, y(L) = 0....Ch. 11.4 - Consider y + y = 0 subject to the periodic...Ch. 11.4 - Prob. 5ECh. 11.4 - Prob. 6ECh. 11.4 - (a) Find the eigenvalues and eigenfunctions of the...Ch. 11.4 - (a) Find the eigenvalues and eigenfunctions of the...Ch. 11.4 - Laguerres differential equation xy + (1 x)y + ny...Ch. 11.4 - Hermites differential equation y2xy+2ny=0,n=0,1,2,...Ch. 11.4 - Consider the regular Sturm-Liouville problem:...Ch. 11.4 - (a) Find the eigenfunctions and the equation that...Ch. 11.4 - Prob. 13ECh. 11.5 - Prob. 1ECh. 11.5 - Prob. 2ECh. 11.5 - Prob. 3ECh. 11.5 - Prob. 4ECh. 11.5 - In Problems 36 expand f(x) = 1, 0 x 2, in a...Ch. 11.5 - In Problems 36 expand f(x) = 1, 0 x 2, in a...Ch. 11.5 - In Problems 7-10 expand the given function in a...Ch. 11.5 - Prob. 8ECh. 11.5 - Prob. 9ECh. 11.5 - Prob. 10ECh. 11.5 - Prob. 13ECh. 11.5 - Prob. 14ECh. 11.5 - In Problems 15 and 16 write out the first five...Ch. 11.5 - Prob. 16ECh. 11.5 - Prob. 17ECh. 11.5 - Prob. 18ECh. 11.5 - Prob. 19ECh. 11.5 - Prob. 20ECh. 11.5 - Prob. 21ECh. 11.5 - Prob. 22ECh. 11.5 - Prob. 23ECh. 11.5 - Prob. 24ECh. 11 - In Problems 16 fill in the blank or answer true or...Ch. 11 - Prob. 2RECh. 11 - Prob. 3RECh. 11 - Prob. 4RECh. 11 - Prob. 5RECh. 11 - Prob. 6RECh. 11 - Prob. 7RECh. 11 - Prob. 8RECh. 11 - Prob. 9RECh. 11 - Prob. 10RECh. 11 - Prob. 11RECh. 11 - Prob. 12RECh. 11 - Prob. 13RECh. 11 - Prob. 14RECh. 11 - Prob. 15RECh. 11 - Prob. 16RECh. 11 - Prob. 17RECh. 11 - Consider the portion of the periodic function f...Ch. 11 - Prob. 19RECh. 11 - Find the eigenvalues and eigenfunctions of the...Ch. 11 - Prob. 21RECh. 11 - Prob. 22RECh. 11 - Prob. 23RECh. 11 - Prob. 24RECh. 11 - Prob. 25RE
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