Calculus (MindTap Course List)
8th Edition
ISBN: 9781285740621
Author: James Stewart
Publisher: Cengage Learning
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Chapter 12.4, Problem 27E
To determine
To find:
The area of the parallelogram with vertices
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(12 points) Let
E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}.
(a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such
that
(x, y, z) (psin cos 0, psin sin 0, p cos) € E.
(b) (8 points) Calculate the integral
E
xyz dV using spherical coordinates.
(10 points) Let f(x, y, z) = ze²²+y². Let
E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}.
Calculate the integral
y,
f(x, y, z) dV.
(14 points) Let f: R3 R and T: R3.
→R³ be defined by
f(x, y, z) = ln(x²+ y²+2²),
T(p, 0,4)=(psin cos 0, psin sin, pcos).
(a) (4 points) Write out the composition g(p, 0, 4) = (foT)(p,, ) explicitly. Then calculate the
gradient Vg directly, i.e. without using the chain rule.
(b) (4 points) Calculate the gradient Vf(x, y, z) where (x, y, z) = T(p, 0,4).
(c) (6 points) Calculate the derivative matrix DT(p, 0, p). Then use the Chain Rule to calculate
Vg(r,0,4).
Chapter 12 Solutions
Calculus (MindTap Course List)
Ch. 12.1 - Suppose you start at the origin, move along the...Ch. 12.1 - Sketch the points (1,5,3),(0,2,3),(3,0,2), and...Ch. 12.1 - Which of the points A(4,0,1),B(3,1,5), and...Ch. 12.1 - What are the projections of the point 2, 3, 5 on...Ch. 12.1 - What does the equation x=4 represent in 2? What...Ch. 12.1 - What does the equation y = 3 represent in 3? What...Ch. 12.1 - Describe and sketch the surface in 3 represented...Ch. 12.1 - Prob. 8ECh. 12.1 - Find the lengths of the sides of the triangle PQR....Ch. 12.1 - Find the lengths of the sides of the triangle PQR....
Ch. 12.1 - Determine whether the points lie on a straight...Ch. 12.1 - Prob. 12ECh. 12.1 - Find an equation of the sphere with centre (3,2,5)...Ch. 12.1 - Find an equation of the sphere with centre (2,6,4)...Ch. 12.1 - Find an equation of the sphere that passes through...Ch. 12.1 - Prob. 16ECh. 12.1 - Prob. 17ECh. 12.1 - Prob. 18ECh. 12.1 - Prob. 19ECh. 12.1 - Prob. 20ECh. 12.1 - a Prove that the midpoint of the line segment from...Ch. 12.1 - Find an equation of a sphere if one of its...Ch. 12.1 - Prob. 23ECh. 12.1 - Find an equation of the largest sphere with centre...Ch. 12.1 - Prob. 25ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 27ECh. 12.1 - Prob. 28ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 30ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 33ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 36ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 38ECh. 12.1 - Prob. 39ECh. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - The figure shows a line L1 in space and a second...Ch. 12.1 - Consider the points P such that the distance from...Ch. 12.1 - Prob. 45ECh. 12.1 - Find the volume of the solid that lies inside both...Ch. 12.1 - Find the distance between the spheres x2+y2+z2=4...Ch. 12.1 - Describe and sketch a solid with the following...Ch. 12.2 - Are the following quantities vectors or scalars?...Ch. 12.2 - What is the relationship between the point 4, 7...Ch. 12.2 - Name all the equal vectors in the parallelogram...Ch. 12.2 - Prob. 4ECh. 12.2 - Prob. 5ECh. 12.2 - Prob. 6ECh. 12.2 - In the figure, the tip of c and the tail of d are...Ch. 12.2 - If the vectors in the figure satisfy |u|=|v|=1 and...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Prob. 10ECh. 12.2 - Prob. 11ECh. 12.2 - Prob. 12ECh. 12.2 - Prob. 13ECh. 12.2 - Prob. 14ECh. 12.2 - Prob. 15ECh. 12.2 - Prob. 16ECh. 12.2 - Prob. 17ECh. 12.2 - Prob. 18ECh. 12.2 - Prob. 19ECh. 12.2 - Find a+b,4a+2b,|a|, and |ab|. a=5i+3j,b=i2jCh. 12.2 - Prob. 21ECh. 12.2 - Find a+b,4a+2b,|a|, and |ab|. a=8,1,4,b=5,2,1Ch. 12.2 - Prob. 23ECh. 12.2 - Prob. 24ECh. 12.2 - Prob. 25ECh. 12.2 - Prob. 26ECh. 12.2 - What is the angle between the given vector and the...Ch. 12.2 - Prob. 28ECh. 12.2 - If v lies in the first quadrant and makes an angle...Ch. 12.2 - Prob. 30ECh. 12.2 - Prob. 31ECh. 12.2 - Find the magnitude of the resultant force and the...Ch. 12.2 - Find the magnitude of the resultant force and the...Ch. 12.2 - Prob. 34ECh. 12.2 - Prob. 35ECh. 12.2 - Prob. 36ECh. 12.2 - A block-and-tackle pulley hoist is suspended in a...Ch. 12.2 - The tension T at each end of a chain has magnitude...Ch. 12.2 - A boatman wants to cross a canal that is 3 km wide...Ch. 12.2 - Three forces act on an object. Two of the forces...Ch. 12.2 - Prob. 41ECh. 12.2 - Prob. 42ECh. 12.2 - If A, B, and C are the vertices of a triangle,...Ch. 12.2 - Prob. 44ECh. 12.2 - a Draw the vectors a=3,2,b=2,1 and c=7,1. b Show,...Ch. 12.2 - Prob. 46ECh. 12.2 - Prob. 47ECh. 12.2 - Prob. 48ECh. 12.2 - Prob. 49ECh. 12.2 - Prove Property 5 of vectors algebraically for the...Ch. 12.2 - Use vectors to prove that the line joining the...Ch. 12.2 - Prob. 52ECh. 12.3 - Which of the following expressions are meaningful?...Ch. 12.3 - Prob. 2ECh. 12.3 - 210 Find ab. a=1.5,0.4, b=4,6Ch. 12.3 - Prob. 4ECh. 12.3 - Prob. 5ECh. 12.3 - Prob. 6ECh. 12.3 - Prob. 7ECh. 12.3 - Prob. 8ECh. 12.3 - Prob. 9ECh. 12.3 - Prob. 10ECh. 12.3 - If u is a unit vector, find uv and uw.Ch. 12.3 - If u is a unit vector, find uv and uw.Ch. 12.3 - Prob. 13ECh. 12.3 - Prob. 14ECh. 12.3 - Prob. 15ECh. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Prob. 17ECh. 12.3 - Prob. 18ECh. 12.3 - Prob. 19ECh. 12.3 - Prob. 20ECh. 12.3 - 2122 Find, correct to the nearest degree, the...Ch. 12.3 - Prob. 22ECh. 12.3 - Prob. 23ECh. 12.3 - 2324 Determine whether the given vectors are...Ch. 12.3 - Prob. 25ECh. 12.3 - Prob. 26ECh. 12.3 - Prob. 27ECh. 12.3 - Prob. 28ECh. 12.3 - Prob. 29ECh. 12.3 - Prob. 30ECh. 12.3 - 3132 Find the acute angles between the curves at...Ch. 12.3 - 3132 Find the acute angles between the curves at...Ch. 12.3 - Prob. 33ECh. 12.3 - Prob. 34ECh. 12.3 - Prob. 35ECh. 12.3 - Prob. 36ECh. 12.3 - Prob. 37ECh. 12.3 - Prob. 38ECh. 12.3 - 3944 Find the scalar and vector projections of b...Ch. 12.3 - Prob. 40ECh. 12.3 - Prob. 41ECh. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Prob. 44ECh. 12.3 - Prob. 45ECh. 12.3 - Prob. 46ECh. 12.3 - If a=3,0,1, find a vector b such that compab=2.Ch. 12.3 - Prob. 48ECh. 12.3 - Prob. 49ECh. 12.3 - Prob. 50ECh. 12.3 - Prob. 51ECh. 12.3 - Prob. 52ECh. 12.3 - Use a scalar projection to show that the distance...Ch. 12.3 - If r=x,y,z,a=a1,a2,a3, and b=b1,b2,b3, show that...Ch. 12.3 - Prob. 55ECh. 12.3 - Prob. 56ECh. 12.3 - Prob. 57ECh. 12.3 - If c=|a|b+|b|a, where a, b, and c are all nonzero...Ch. 12.3 - Prob. 59ECh. 12.3 - Prob. 60ECh. 12.3 - Use Theorem 3 to prove the Cauchy-Schwarz...Ch. 12.3 - Prob. 62ECh. 12.3 - The Parallelogram Law states that...Ch. 12.3 - Prob. 64ECh. 12.3 - Prob. 65ECh. 12.4 - Prob. 1ECh. 12.4 - Prob. 2ECh. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - Prob. 5ECh. 12.4 - Prob. 6ECh. 12.4 - Prob. 7ECh. 12.4 - If a=i2k and b=j+k, find ab. Sketch a, b, and ab....Ch. 12.4 - Prob. 9ECh. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - Prob. 13ECh. 12.4 - Prob. 14ECh. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - Prob. 17ECh. 12.4 - If a=1,0,1,b=2,1,1, and c=0,1,3, show that...Ch. 12.4 - Find two unit vectors orthogonal to both 3,2,1 and...Ch. 12.4 - Prob. 20ECh. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - Prob. 25ECh. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Find the area of the parallelogram with vertices...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - Prob. 32ECh. 12.4 - Prob. 33ECh. 12.4 - Prob. 34ECh. 12.4 - Prob. 35ECh. 12.4 - Prob. 36ECh. 12.4 - Prob. 37ECh. 12.4 - Prob. 38ECh. 12.4 - Prob. 39ECh. 12.4 - Prob. 40ECh. 12.4 - A wrench 30 cm long lies along the positive y-axis...Ch. 12.4 - Prob. 42ECh. 12.4 - Prob. 43ECh. 12.4 - a Find all vectors v such that 1,2,1v=3,1,5 b...Ch. 12.4 - Prob. 45ECh. 12.4 - a Let P be a point not on the plane that passes...Ch. 12.4 - Prob. 47ECh. 12.4 - Prob. 48ECh. 12.4 - Prob. 49ECh. 12.4 - Prob. 50ECh. 12.4 - Prob. 51ECh. 12.4 - Prob. 52ECh. 12.4 - Suppose that a0. a If ab=ac, does it follow that...Ch. 12.4 - If v1, v2, and v3 are noncoplanar vectors, let...Ch. 12.5 - Determine whether each statement is true or false...Ch. 12.5 - Prob. 2ECh. 12.5 - Prob. 3ECh. 12.5 - Prob. 4ECh. 12.5 - Prob. 5ECh. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - Prob. 7ECh. 12.5 - Prob. 8ECh. 12.5 - Prob. 9ECh. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - Prob. 13ECh. 12.5 - Is the line through (2,4,0) and (1,1,1)...Ch. 12.5 - a Find symmetric equations for the line that...Ch. 12.5 - a Find parametric equations for the line through...Ch. 12.5 - Prob. 17ECh. 12.5 - Prob. 18ECh. 12.5 - 1922 Determine whether the lines L1 and L2 are...Ch. 12.5 - Prob. 20ECh. 12.5 - Prob. 21ECh. 12.5 - Prob. 22ECh. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - Prob. 40ECh. 12.5 - Prob. 41ECh. 12.5 - Prob. 42ECh. 12.5 - Prob. 43ECh. 12.5 - Prob. 44ECh. 12.5 - 4547 Find the point at which the line intersects...Ch. 12.5 - Prob. 46ECh. 12.5 - Prob. 47ECh. 12.5 - Prob. 48ECh. 12.5 - Find direction numbers for the line of...Ch. 12.5 - Prob. 50ECh. 12.5 - Prob. 51ECh. 12.5 - Prob. 52ECh. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - Prob. 57ECh. 12.5 - Prob. 58ECh. 12.5 - Prob. 59ECh. 12.5 - 5960 Find symmetric equations for the line of...Ch. 12.5 - Prob. 61ECh. 12.5 - Prob. 62ECh. 12.5 - Prob. 63ECh. 12.5 - Prob. 64ECh. 12.5 - Prob. 65ECh. 12.5 - Prob. 66ECh. 12.5 - Prob. 67ECh. 12.5 - Prob. 68ECh. 12.5 - Prob. 69ECh. 12.5 - Prob. 70ECh. 12.5 - 7172 Find the distance from the point to the given...Ch. 12.5 - Prob. 72ECh. 12.5 - Prob. 73ECh. 12.5 - Prob. 74ECh. 12.5 - Prob. 75ECh. 12.5 - Prob. 76ECh. 12.5 - Show that the lines with symmetric equations x=y=z...Ch. 12.5 - Prob. 78ECh. 12.5 - Prob. 79ECh. 12.5 - Prob. 80ECh. 12.5 - Prob. 81ECh. 12.5 - Prob. 82ECh. 12.5 - If a, b, and c are not all 0, show that the...Ch. 12.6 - a What does the equation y=x2 represent as a curve...Ch. 12.6 - a Sketch the graph of y=ex as a curve in 2. b...Ch. 12.6 - 38 Describe and sketch the surface. x2+z2=1Ch. 12.6 - 38 Describe and sketch the surface. 4x2+y2=4Ch. 12.6 - 38 Describe and sketch the surface. z=1y2Ch. 12.6 - 38 Describe and sketch the surface. y=z2Ch. 12.6 - 38 Describe and sketch the surface. xy=1Ch. 12.6 - 38 Describe and sketch the surface. z=sinyCh. 12.6 - a Find and identify the traces of the quadric...Ch. 12.6 - a Find and identify the traces of the quadric...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - Prob. 24ECh. 12.6 - Prob. 25ECh. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - Prob. 28ECh. 12.6 - 2930 Sketch and identify a quadric surface that...Ch. 12.6 - Prob. 30ECh. 12.6 - Prob. 31ECh. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - Prob. 33ECh. 12.6 - Prob. 34ECh. 12.6 - Prob. 35ECh. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - Prob. 37ECh. 12.6 - Prob. 38ECh. 12.6 - Prob. 39ECh. 12.6 - Prob. 40ECh. 12.6 - Prob. 41ECh. 12.6 - Prob. 42ECh. 12.6 - Prob. 43ECh. 12.6 - Prob. 44ECh. 12.6 - Find an equation for the surface obtained by...Ch. 12.6 - Prob. 46ECh. 12.6 - Prob. 47ECh. 12.6 - Find an equation for the surface consisting of all...Ch. 12.6 - Traditionally, the earths surface has been modeled...Ch. 12.6 - A cooling tower for a nuclear reactor is to be...Ch. 12.6 - Show that if the point (a,b,c) lies on the...Ch. 12.6 - Show that the curve of intersection of the...Ch. 12.6 - Prob. 53ECh. 12.R - What is the difference between a vector and a...Ch. 12.R - Prob. 2CCCh. 12.R - Prob. 3CCCh. 12.R - Prob. 4CCCh. 12.R - Prob. 5CCCh. 12.R - Prob. 6CCCh. 12.R - Prob. 7CCCh. 12.R - Prob. 8CCCh. 12.R - Prob. 9CCCh. 12.R - Prob. 10CCCh. 12.R - Prob. 11CCCh. 12.R - Prob. 12CCCh. 12.R - Prob. 13CCCh. 12.R - Prob. 14CCCh. 12.R - Prob. 15CCCh. 12.R - Prob. 16CCCh. 12.R - Prob. 17CCCh. 12.R - What are the traces of a surface? How do you find...Ch. 12.R - Prob. 19CCCh. 12.R - Prob. 1TFQCh. 12.R - Prob. 2TFQCh. 12.R - Prob. 3TFQCh. 12.R - Prob. 4TFQCh. 12.R - Prob. 5TFQCh. 12.R - Prob. 6TFQCh. 12.R - Prob. 7TFQCh. 12.R - Prob. 8TFQCh. 12.R - Prob. 9TFQCh. 12.R - Prob. 10TFQCh. 12.R - Prob. 11TFQCh. 12.R - Prob. 12TFQCh. 12.R - Prob. 13TFQCh. 12.R - Prob. 14TFQCh. 12.R - Prob. 15TFQCh. 12.R - Prob. 16TFQCh. 12.R - Prob. 17TFQCh. 12.R - Prob. 18TFQCh. 12.R - Prob. 19TFQCh. 12.R - Prob. 20TFQCh. 12.R - Prob. 21TFQCh. 12.R - Prob. 22TFQCh. 12.R - Prob. 1ECh. 12.R - Prob. 2ECh. 12.R - Prob. 3ECh. 12.R - Calculate the given quantity if...Ch. 12.R - Prob. 5ECh. 12.R - Prob. 6ECh. 12.R - Prob. 7ECh. 12.R - Prob. 8ECh. 12.R - Prob. 9ECh. 12.R - Prob. 10ECh. 12.R - Prob. 11ECh. 12.R - Prob. 12ECh. 12.R - Prob. 13ECh. 12.R - Prob. 14ECh. 12.R - Prob. 15ECh. 12.R - Prob. 16ECh. 12.R - Prob. 17ECh. 12.R - Prob. 18ECh. 12.R - Prob. 19ECh. 12.R - Prob. 20ECh. 12.R - Find the point in which the line with parametric...Ch. 12.R - Prob. 22ECh. 12.R - Prob. 23ECh. 12.R - a Show that the planes x+yz=1 and 2x3y+4z=5 are...Ch. 12.R - Prob. 25ECh. 12.R - a Find an equation of the plane that passes...Ch. 12.R - Prob. 27ECh. 12.R - Prob. 28ECh. 12.R - Prob. 29ECh. 12.R - Prob. 30ECh. 12.R - Prob. 31ECh. 12.R - Prob. 32ECh. 12.R - Prob. 33ECh. 12.R - Prob. 34ECh. 12.R - Prob. 35ECh. 12.R - Prob. 36ECh. 12.R - Prob. 37ECh. 12.R - Prob. 38ECh. 12.P - Each edge of a cubical box has length 1 m. The box...Ch. 12.P - Let B be a solid box with length L, width W, and...Ch. 12.P - Let L be the line of intersection of the planes...Ch. 12.P - A plane is capable of flying at a speed of 180...Ch. 12.P - Prob. 5PCh. 12.P - Find an equation of the largest sphere that passes...Ch. 12.P - Prob. 7PCh. 12.P - A solid has the following properties. When...
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