(a) Explanation Given information: The hyperboloid of one sheet x 2 + y 2 − z 2 = 1 , and θ associated with the circle x 2 + y 2 = r 2 and the hyperbolic parameter u associated with the hyperbolic function r 2 − z 2 = 1 . Calculation: Write the Equation of hyperboloid. x 2 + y 2 − z 2 = 1 (1) The hyperbolic parameter u associated with the hyperbolic function. Take r = cosh u and z = sinh u . Then r 2 − z 2 = 1 ( cosh 2 u − sinh 2 u = 1 ) . The equation of circle is x 2 + y 2 = r 2 . The parametrization for the circle is r ( r , θ ) = x i + y j + z k = ( r cos θ ) i + ( r sin θ ) j + z k (2) From the hyperbolic function and Equation (2), x = cosh u cos θ , y = cosh u sin θ and z = sinh u (b) To determine Generalize the result in part (a) to the hyperboloid ( x 2 a 2 ) + ( y 2 a 2 ) − ( z 2 a 2 ) = 1 .

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Chapter 16.5, Problem 34E

(a)

To determine

Find the parametrization for the hyperboloid of one sheet x2+y2−z2=1, and θ associated with the circle x2+y2=r2 and the hyperbolic parameter u associated with the hyperbolic function r2−z2=1.

(b)

To determine

Generalize the result in part (a) to the hyperboloid (x2a2)+(y2a2)−(z2a2)=1.

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Chapter 16.5, Problem 33E

Chapter 16.5, Problem 35E

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Chapter 16 Solutions

Pearson eText for Thomas' Calculus: Early Transcendentals -- Instant Access (Pearson+)

Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Evaluate ∫C (x + y) ds, where C is the...Ch. 16.1 - Prob. 10E

Ch. 16.1 - Evaluate ∫C (xy + y + z) ds along the curve r(t) =...Ch. 16.1 - Evaluate along the curve r(t) = (4 cos t)i + (4...Ch. 16.1 - Find the line integral of f(x, y, z) = x + y + z...Ch. 16.1 - Prob. 14E Ch. 16.1 - Prob. 15E Ch. 16.1 - Integrate over the path C1 followed by C2...Ch. 16.1 - Prob. 17E Ch. 16.1 - Prob. 18E Ch. 16.1 - Evaluate ∫C x ds, where C is the straight-line...Ch. 16.1 - Evaluate , where C is the straight-line segment x...Ch. 16.1 - Prob. 21E Ch. 16.1 - Find the line integral of f(x, y) = x − y + 3...Ch. 16.1 - Prob. 23E Ch. 16.1 - Prob. 24E Ch. 16.1 - Prob. 25E Ch. 16.1 - Evaluate , where C is given in the accompanying...Ch. 16.1 - Prob. 27E Ch. 16.1 - Prob. 28E Ch. 16.1 - Prob. 29E Ch. 16.1 - Prob. 30E Ch. 16.1 - Find the area of one side of the “winding wall”...Ch. 16.1 - Prob. 32E Ch. 16.1 - Prob. 33E Ch. 16.1 - Center of mass of a curved wire A wire of density ...Ch. 16.1 - Prob. 35E Ch. 16.1 - Prob. 36E Ch. 16.1 - Prob. 37E Ch. 16.1 - Prob. 38E Ch. 16.1 - Prob. 39E Ch. 16.1 - Prob. 40E Ch. 16.1 - Prob. 41E Ch. 16.1 - Prob. 42E Ch. 16.2 - Find the gradient fields of the functions in...Ch. 16.2 - Prob. 2E Ch. 16.2 - Prob. 3E Ch. 16.2 - Prob. 4E Ch. 16.2 - Prob. 5E Ch. 16.2 - Prob. 6E Ch. 16.2 - In Exercises 7−12, find the line integrals of F...Ch. 16.2 - Prob. 8E Ch. 16.2 - Prob. 9E Ch. 16.2 - Prob. 10E Ch. 16.2 - Line Integrals of Vector Fields In Exercises 7−12,...Ch. 16.2 - Prob. 12E Ch. 16.2 - Prob. 13E Ch. 16.2 - Prob. 14E Ch. 16.2 - In Exercises 13–16, find the line integrals along...Ch. 16.2 - Prob. 16E Ch. 16.2 - Prob. 17E Ch. 16.2 - Prob. 18E Ch. 16.2 - In Exercises 19–22, find the work done by F over...Ch. 16.2 - Prob. 20E Ch. 16.2 - Prob. 21E Ch. 16.2 - Prob. 22E Ch. 16.2 - Prob. 23E Ch. 16.2 - Prob. 24E Ch. 16.2 - Prob. 25E Ch. 16.2 - Prob. 26E Ch. 16.2 - Prob. 27E Ch. 16.2 - Prob. 28E Ch. 16.2 - Prob. 29E Ch. 16.2 - Prob. 30E Ch. 16.2 - Prob. 31E Ch. 16.2 - Prob. 32E Ch. 16.2 - In Exercises 31–34, find the circulation and flux...Ch. 16.2 - Prob. 34E Ch. 16.2 - Prob. 35E Ch. 16.2 - Prob. 36E Ch. 16.2 - Prob. 37E Ch. 16.2 - Prob. 38E Ch. 16.2 - Prob. 39E Ch. 16.2 - Find the circulation of the field F = yi + (x +...Ch. 16.2 - Prob. 41E Ch. 16.2 - Prob. 42E Ch. 16.2 - Prob. 43E Ch. 16.2 - Prob. 44E Ch. 16.2 - Prob. 45E Ch. 16.2 - Prob. 46E Ch. 16.2 - Prob. 47E Ch. 16.2 - Prob. 48E Ch. 16.2 - A field of tangent vectors Find a field G = P(x,...Ch. 16.2 - Prob. 50E Ch. 16.2 - Prob. 51E Ch. 16.2 - Prob. 52E Ch. 16.2 - Prob. 53E Ch. 16.2 - Work done by a radial force with constant...Ch. 16.2 - Prob. 55E Ch. 16.2 - Prob. 56E Ch. 16.2 - Prob. 57E Ch. 16.2 - Prob. 58E Ch. 16.2 - Circulation Find the circulation of F = 2xi + 2zj...Ch. 16.2 - Prob. 60E Ch. 16.2 - Prob. 61E Ch. 16.2 - Prob. 62E Ch. 16.3 - Which fields in Exercises 1–6 are conservative,...Ch. 16.3 - Prob. 2E Ch. 16.3 - Prob. 3E Ch. 16.3 - Prob. 4E Ch. 16.3 - Prob. 5E Ch. 16.3 - Prob. 6E Ch. 16.3 - Finding Potential Functions In Exercises 7–12,...Ch. 16.3 - In Exercises 7–12, find a potential function f...Ch. 16.3 - In Exercises 7–12, find a potential function f for...Ch. 16.3 - Prob. 10E Ch. 16.3 - In Exercises 7–12, find a potential function f for...Ch. 16.3 - Prob. 12E Ch. 16.3 - Prob. 13E Ch. 16.3 - Prob. 14E Ch. 16.3 - Prob. 15E Ch. 16.3 - Prob. 16E Ch. 16.3 - Prob. 17E Ch. 16.3 - Prob. 18E Ch. 16.3 - Prob. 19E Ch. 16.3 - Prob. 20E Ch. 16.3 - Prob. 21E Ch. 16.3 - Prob. 22E Ch. 16.3 - Prob. 23E Ch. 16.3 - Prob. 24E Ch. 16.3 - Prob. 25E Ch. 16.3 - Prob. 26E Ch. 16.3 - Prob. 27E Ch. 16.3 - Prob. 28E Ch. 16.3 - Work along different paths Find the work done by F...Ch. 16.3 - Prob. 30E Ch. 16.3 - Prob. 31E Ch. 16.3 - Integral along different paths Evaluate the line...Ch. 16.3 - Prob. 33E Ch. 16.3 - Prob. 34E Ch. 16.3 - Prob. 35E Ch. 16.3 - Prob. 36E Ch. 16.3 - Prob. 37E Ch. 16.3 - Gravitational field Find a potential function for...Ch. 16.4 - In Exercises 1–6, find the k-component of curl(F)...Ch. 16.4 - Prob. 2E Ch. 16.4 - Prob. 3E Ch. 16.4 - Prob. 4E Ch. 16.4 - Prob. 5E Ch. 16.4 - Prob. 6E Ch. 16.4 - Prob. 7E Ch. 16.4 - In Exercises 7–10, verify the conclusion of...Ch. 16.4 - Prob. 9E Ch. 16.4 - Prob. 10E Ch. 16.4 - Prob. 11E Ch. 16.4 - Prob. 12E Ch. 16.4 - Prob. 13E Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - Prob. 15E Ch. 16.4 - Prob. 16E Ch. 16.4 - Prob. 17E Ch. 16.4 - Prob. 18E Ch. 16.4 - Prob. 19E Ch. 16.4 - Prob. 20E Ch. 16.4 - Prob. 21E Ch. 16.4 - Prob. 22E Ch. 16.4 - Prob. 23E Ch. 16.4 - Prob. 24E Ch. 16.4 - Prob. 25E Ch. 16.4 - Prob. 26E Ch. 16.4 - Prob. 27E Ch. 16.4 - Prob. 28E Ch. 16.4 - Prob. 29E Ch. 16.4 - Prob. 30E Ch. 16.4 - Prob. 31E Ch. 16.4 - Use the Green’s Theorem area formula given above...Ch. 16.4 - Use the Green’s Theorem area formula given above...Ch. 16.4 - Use the Green’s Theorem area formula given above...Ch. 16.4 - Prob. 35E Ch. 16.4 - Prob. 36E Ch. 16.4 - Prob. 37E Ch. 16.4 - Prob. 38E Ch. 16.4 - Prob. 39E Ch. 16.4 - Prob. 40E Ch. 16.4 - Prob. 41E Ch. 16.4 - Prob. 42E Ch. 16.4 - Prob. 43E Ch. 16.4 - Prob. 44E Ch. 16.4 - Prob. 45E Ch. 16.4 - Prob. 46E Ch. 16.4 - Prob. 47E Ch. 16.4 - Prob. 48E Ch. 16.5 - In Exercises 1–16, find a parametrization of the...Ch. 16.5 - Prob. 2E Ch. 16.5 - Prob. 3E Ch. 16.5 - Prob. 4E Ch. 16.5 - Prob. 5E Ch. 16.5 - Prob. 6E Ch. 16.5 - Prob. 7E Ch. 16.5 - Prob. 8E Ch. 16.5 - Prob. 9E Ch. 16.5 - Prob. 10E Ch. 16.5 - Prob. 11E Ch. 16.5 - Prob. 12E Ch. 16.5 - Prob. 13E Ch. 16.5 - Prob. 14E Ch. 16.5 - Prob. 15E Ch. 16.5 - Prob. 16E Ch. 16.5 - Prob. 17E Ch. 16.5 - Prob. 18E Ch. 16.5 - Prob. 19E Ch. 16.5 - Prob. 20E Ch. 16.5 - In Exercises 17–26, use a parametrization to...Ch. 16.5 - In Exercises 17–26, use a parametrization to...Ch. 16.5 - Prob. 23E Ch. 16.5 - Prob. 24E Ch. 16.5 - Prob. 25E Ch. 16.5 - Prob. 26E Ch. 16.5 - Prob. 27E Ch. 16.5 - Prob. 28E Ch. 16.5 - Prob. 29E Ch. 16.5 - Prob. 30E Ch. 16.5 - Prob. 31E Ch. 16.5 - Prob. 32E Ch. 16.5 - Prob. 33E Ch. 16.5 - Prob. 34E Ch. 16.5 - Prob. 35E Ch. 16.5 - Prob. 36E Ch. 16.5 - Prob. 37E Ch. 16.5 - Prob. 38E Ch. 16.5 - Prob. 39E Ch. 16.5 - Prob. 40E Ch. 16.5 - Prob. 41E Ch. 16.5 - Find the area of the cap cut from the sphere x2 +...Ch. 16.5 - Prob. 43E Ch. 16.5 - Prob. 44E Ch. 16.5 - Prob. 45E Ch. 16.5 - Prob. 46E Ch. 16.5 - Prob. 47E Ch. 16.5 - Prob. 48E Ch. 16.5 - Prob. 49E Ch. 16.5 - Prob. 50E Ch. 16.5 - Prob. 51E Ch. 16.5 - Find the area of the surfaces in Exercises...Ch. 16.5 - Prob. 53E Ch. 16.5 - Prob. 54E Ch. 16.5 - Prob. 55E Ch. 16.5 - Prob. 56E Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - Prob. 5E Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - Prob. 7E Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - Prob. 9E Ch. 16.6 - Prob. 10E Ch. 16.6 - Prob. 11E Ch. 16.6 - Prob. 12E Ch. 16.6 - Prob. 13E Ch. 16.6 - Prob. 14E Ch. 16.6 - Prob. 15E Ch. 16.6 - Integrate G(x, y, z) = x over the surface given by...Ch. 16.6 - Prob. 17E Ch. 16.6 - Integrate G(x, y, z) = x – y – z over the portion...Ch. 16.6 - Prob. 19E Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - Prob. 21E Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - Prob. 25E Ch. 16.6 - Prob. 26E Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - Prob. 28E Ch. 16.6 - Prob. 29E Ch. 16.6 - Prob. 30E Ch. 16.6 - Prob. 31E Ch. 16.6 - Prob. 32E Ch. 16.6 - Prob. 33E Ch. 16.6 - Prob. 34E Ch. 16.6 - Prob. 35E Ch. 16.6 - Prob. 36E Ch. 16.6 - Prob. 37E Ch. 16.6 - Prob. 38E Ch. 16.6 - Prob. 39E Ch. 16.6 - Prob. 40E Ch. 16.6 - Prob. 41E Ch. 16.6 - Prob. 42E Ch. 16.6 - Prob. 43E Ch. 16.6 - Prob. 44E Ch. 16.6 - Prob. 45E Ch. 16.6 - Prob. 46E Ch. 16.6 - Prob. 47E Ch. 16.6 - Prob. 48E Ch. 16.6 - Prob. 49E Ch. 16.6 - Prob. 50E Ch. 16.7 - Prob. 1E Ch. 16.7 - Prob. 2E Ch. 16.7 - Prob. 3E Ch. 16.7 - Prob. 4E Ch. 16.7 - Prob. 5E Ch. 16.7 - Prob. 6E Ch. 16.7 - Prob. 7E Ch. 16.7 - Prob. 8E Ch. 16.7 - Prob. 9E Ch. 16.7 - In Exercises 7–12, use the surface integral in...Ch. 16.7 - Prob. 11E Ch. 16.7 - Prob. 12E Ch. 16.7 - Prob. 13E Ch. 16.7 - Prob. 14E Ch. 16.7 - Prob. 15E Ch. 16.7 - Evaluate where S is the hemisphere x2 + y2 + z2 =...Ch. 16.7 - Prob. 17E Ch. 16.7 - Prob. 18E Ch. 16.7 - In Exercises 19–24, use the surface integral in...Ch. 16.7 - Prob. 20E Ch. 16.7 - In Exercises 19–24, use the surface integral in...Ch. 16.7 - Prob. 22E Ch. 16.7 - Prob. 23E Ch. 16.7 - Prob. 24E Ch. 16.7 - Prob. 25E Ch. 16.7 - Verify Stokes’ Theorem for the vector field F =...Ch. 16.7 - Prob. 27E Ch. 16.7 - Prob. 28E Ch. 16.7 - Prob. 29E Ch. 16.7 - Prob. 30E Ch. 16.7 - Prob. 31E Ch. 16.7 - Does Stokes’ Theorem say anything special about...Ch. 16.7 - Prob. 33E Ch. 16.7 - Prob. 34E Ch. 16.8 - In Exercises 1–8, find the divergence of the...Ch. 16.8 - Prob. 2E Ch. 16.8 - In Exercises 1–8, find the divergence of the...Ch. 16.8 - Prob. 4E Ch. 16.8 - Prob. 5E Ch. 16.8 - Prob. 6E Ch. 16.8 - Prob. 7E Ch. 16.8 - In Exercises 1–8, find the divergence of the...Ch. 16.8 - Prob. 9E Ch. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - Prob. 11E Ch. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - Prob. 13E Ch. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - Prob. 15E Ch. 16.8 - Prob. 16E Ch. 16.8 - Prob. 17E Ch. 16.8 - Prob. 18E Ch. 16.8 - Prob. 19E Ch. 16.8 - Prob. 20E Ch. 16.8 - Prob. 21E Ch. 16.8 - Prob. 22E Ch. 16.8 - Prob. 23E Ch. 16.8 - Prob. 24E Ch. 16.8 - Prob. 25E Ch. 16.8 - Prob. 26E Ch. 16.8 - Calculate the net outward flux of the vector...Ch. 16.8 - Prob. 28E Ch. 16.8 - Prob. 29E Ch. 16.8 - Prob. 30E Ch. 16.8 - Prob. 31E Ch. 16.8 - Prob. 32E Ch. 16.8 - Prob. 33E Ch. 16.8 - Green’s second formula (Continuation of Exercise...Ch. 16.8 - Prob. 35E Ch. 16.8 - Prob. 36E Ch. 16 - Prob. 1GYR Ch. 16 - Prob. 2GYR Ch. 16 - Prob. 3GYR Ch. 16 - Prob. 4GYR Ch. 16 - Prob. 5GYR Ch. 16 - Prob. 6GYR Ch. 16 - Prob. 7GYR Ch. 16 - Prob. 8GYR Ch. 16 - Prob. 9GYR Ch. 16 - Prob. 10GYR Ch. 16 - Prob. 11GYR Ch. 16 - Prob. 12GYR Ch. 16 - Prob. 13GYR Ch. 16 - Prob. 14GYR Ch. 16 - Prob. 15GYR Ch. 16 - Prob. 16GYR Ch. 16 - Prob. 17GYR Ch. 16 - Prob. 18GYR Ch. 16 - Prob. 1PE Ch. 16 - Prob. 2PE Ch. 16 - Prob. 3PE Ch. 16 - Prob. 4PE Ch. 16 - Prob. 5PE Ch. 16 - Prob. 6PE Ch. 16 - Prob. 7PE Ch. 16 - Prob. 8PE Ch. 16 - Prob. 9PE Ch. 16 - Prob. 10PE Ch. 16 - Prob. 11PE Ch. 16 - Area of a parabolic cap Find the area of the cap...Ch. 16 - Prob. 13PE Ch. 16 - Prob. 14PE Ch. 16 - Prob. 15PE Ch. 16 - Prob. 16PE Ch. 16 - Prob. 17PE Ch. 16 - Prob. 18PE Ch. 16 - Prob. 19PE Ch. 16 - Prob. 20PE Ch. 16 - Prob. 21PE Ch. 16 - Prob. 22PE Ch. 16 - Prob. 23PE Ch. 16 - Prob. 24PE Ch. 16 - Prob. 25PE Ch. 16 - Prob. 26PE Ch. 16 - Prob. 27PE Ch. 16 - Prob. 28PE Ch. 16 - Prob. 29PE Ch. 16 - Prob. 30PE Ch. 16 - Prob. 31PE Ch. 16 - Prob. 32PE Ch. 16 - Prob. 33PE Ch. 16 - Find potential functions for the fields in...Ch. 16 - Prob. 35PE Ch. 16 - Prob. 36PE Ch. 16 - Prob. 37PE Ch. 16 - Prob. 38PE Ch. 16 - Prob. 39PE Ch. 16 - Prob. 40PE Ch. 16 - Prob. 41PE Ch. 16 - Prob. 42PE Ch. 16 - Prob. 43PE Ch. 16 - Prob. 44PE Ch. 16 - Prob. 45PE Ch. 16 - Prob. 46PE Ch. 16 - Prob. 47PE Ch. 16 - Prob. 48PE Ch. 16 - Prob. 49PE Ch. 16 - Prob. 50PE Ch. 16 - Prob. 51PE Ch. 16 - Prob. 52PE Ch. 16 - Prob. 53PE Ch. 16 - Prob. 54PE Ch. 16 - Prob. 55PE Ch. 16 - Prob. 56PE Ch. 16 - Prob. 57PE Ch. 16 - Prob. 58PE Ch. 16 - Prob. 59PE Ch. 16 - Prob. 60PE Ch. 16 - Prob. 1AAE Ch. 16 - Prob. 2AAE Ch. 16 - Prob. 3AAE Ch. 16 - Prob. 4AAE Ch. 16 - Prob. 5AAE Ch. 16 - Prob. 6AAE Ch. 16 - Prob. 7AAE Ch. 16 - Find the mass of a helicoids r(r, ) = (r cos )i +...Ch. 16 - Prob. 9AAE Ch. 16 - Prob. 10AAE Ch. 16 - Prob. 11AAE Ch. 16 - Prob. 12AAE Ch. 16 - Archimedes’ principle If an object such as a ball...Ch. 16 - Prob. 14AAE Ch. 16 - Prob. 15AAE Ch. 16 - Prob. 16AAE Ch. 16 - Prob. 17AAE Ch. 16 - Prob. 18AAE Ch. 16 - Prob. 19AAE Ch. 16 - Prob. 20AAE Ch. 16 - 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Find the parametrization for the hyperboloid of one sheet x 2 + y 2 − z 2 = 1 , and θ associated with the circle x 2 + y 2 = r 2 and the hyperbolic parameter u associated with the hyperbolic function r 2 − z 2 = 1 .