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Calculus

10th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285057095

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Section
BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285057095
Chapter 15.5, Problem 37E
Textbook Problem
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Finding a Tangent Plane In Exercises 33-36, find an equation of the tangent plane to the surfacerepresented by the vector-valued function at thegiven point.

r ( u , v ) = 2 u cos v i + 3 u sin v j + u 2 k , ( 0 , 6 , 4 )

Chapter 15.5, Problem 37E, Finding a Tangent Plane In Exercises 33-36, find an equation of the tangent plane to the

To determine

To calculate: An equation of the tangent plane to the surface represented by vector-valued function r(u,v)=2ucosvi+3usinvj+u2k at the point (0,6,4).

Explanation of Solution

Given:

The vector valued function is r(u,v)=2ucosvi+3usinvj+u2k and the point (0,6,4).

The graph of the surface is,

Formula used:

The equation of parametric surface,

r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k

The parametric equation is,

x=x(u,v)y=y(u,v)z=z(u,v)

The cross product of the two vectors A=a1i+a2j+a3k and B=b1i+b2j+b3k is,

ijka1a2a3b1b2b3=(a2b3b2a3)i(a1b3b1a3)j+(a1b2b1a2)k

Calculation:

Consider the vector value function, r(u,v)=2ucosvi+3usinvj+u2k.

The equation of parametric surface in standard form is,

r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k

Now, compare the parametric equation of surface with provided vector valued function that gives a parametric equation,

x(u,v)=2ucosvy(u,v)=3usinvz(u,v)=u2

Divide y by x on both side of the above equation to get,

yx=3usinv2ucosvyx=32tanvv=arctan(2y3x)

At the given point x=0,y=6,

Therefore,

v=arctan(2630)v=arctanv=π2

As,

z=u2u=z

At the given point, z=4.

Therefore,

u=2

The point in the uv-plane that is mapped to the point (x,y,z)=(0,6,4) is (u,v)=(2,π2)

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Chapter 15 Solutions

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Orientable Surface...Ch. 15.6 - 57095-15.6-41E-Question-Digital.docx WRITING ABOUT...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 1–4,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 9 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 5 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 11 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 11 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 7 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 7 and...Ch. 15.6 - Mass In Exercises 13 and 14, find the mass of the...Ch. 15.6 - Mass In Exercises 13 and 14, find the mass of the...Ch. 15.6 - Evaluating a Surface Integral In Exercises15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 15-18,...Ch. 15.6 - 57095-15.6-15E-Question-Digital.docx Evaluating a...Ch. 15.6 - Evaluating a Surface Integral In Exercises 15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 31 and 32,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 31 and 32,...Ch. 15.6 - Flow Rate In Exercises 33 and 34, use a computer...Ch. 15.6 - Flow Rate In Exercises 33-34, use a computer...Ch. 15.6 - 57095-15.6-31E-Question-Digital.docx Electrical...Ch. 15.6 - 32. Electrical Charge Let E = xi + yj + 2zk be an...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - HOW DO YOU SEE IT? Is the surface shown in the...Ch. 15.6 - Investigation (a) Use a computer algebra system to...Ch. 15.7 - Classifying a Point in a Vector Field How do you...Ch. 15.7 - 57095-15.7-1E-Question-Digital.docx Verifying the...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 1–6,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - 57095-15.7-17E-Question-Digital.docx Using the...Ch. 15.7 - Using the Divergence Theorem In Exercises 17 and...Ch. 15.7 - 57095-15.7-19E-Question-Digital.docx WRITING ABOUT...Ch. 15.7 - EXPLORING CONCEPTS Closed Surface What is the...Ch. 15.7 - HOW DO YOU SEE IT? The graph of a vector field F...Ch. 15.7 - Volume (a) Use the Divergence Theorem to verify...Ch. 15.7 - Constant Vector Field For the constant vector...Ch. 15.7 - Volume For the vector field F(x,y,z)=xi+yj+zk,...Ch. 15.7 - Verifying an Identity For the vector field...Ch. 15.7 - Proof In Exercises 31 and 32, prove the identity,...Ch. 15.7 - Proof In Exercises 31 and 32, prove the identity,...Ch. 15.8 - CONCEPT CHECK Stokess Theorem Explain the benefit...Ch. 15.8 - Curl What is the physical interpretation of curl?Ch. 15.8 - Finding the Curl of a Vector Field In Exercises...Ch. 15.8 - Finding the Curl of a Vector Field In Exercises...Ch. 15.8 - Finding the Curl of a Vector Field In Exercises...Ch. 15.8 - Finding the Curl of a Vector Field In Exercises...Ch. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokes Theorem In Exercises 3-6, verify...Ch. 15.8 - Using Stokess Theorem In Exercises 716, use...Ch. 15.8 - Using Stokes's Theorem In Exercises 9–18, use...Ch. 15.8 - Using Stokess Theorem In Exercises 716, use...Ch. 15.8 - Using Stokess TheoremIn Exercises 716, use Stokess...Ch. 15.8 - Using Stokess TheoremIn Exercises 716, use Stokess...Ch. 15.8 - Using Stokess TheoremIn Exercises 716, use Stokess...Ch. 15.8 - Using Stokess Theorem In Exercises 7-16, use...Ch. 15.8 - Using Stokess Theorem In Exercises 7-16, use...Ch. 15.8 - Using Stokes Theorem In Exercises 7-16, use Stokes...Ch. 15.8 - Using Stokes Theorem In Exercises 7-16, use Stokes...Ch. 15.8 - Motion of a Liquid In Exercises 19 and 20, the...Ch. 15.8 - Motion of a Liquid In Exercises 19 and 20, the...Ch. 15.8 - 23. Proof Let C be a constant vector. Let S be an...Ch. 15.8 - HOW DO YOU SEE IT? Let S1 be the portion of the...Ch. 15.8 - Let G(x,y)=(yx2+4y2,xx2+4y2,0). Prove or disprove...Ch. 15 - Sketching a Vector Field In Exercises 1 and 2,...Ch. 15 - Sketching a Vector Field In Exercises 1 and 2,...Ch. 15 - Finding a Conservative Vector Field In Exercises...Ch. 15 - Finding a Conservative Vector Field In Exercises...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral In Exercises 21–26,...Ch. 15 - 57095-15-26RE-Question-Digital.docx Evaluating a...Ch. 15 - Evaluating a Line Integral Using Technology In...Ch. 15 - Evaluating a Line Integral Using Technology In...Ch. 15 - Lateral Surface Area In Exercises 43 and44, find...Ch. 15 - Lateral Surface Area In Exercises 43 and44, find...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - 57095-15-35RE-Question-Digital.docx Evaluating a...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - 57095-15-37RE-Question-Digital.docx Evaluating a...Ch. 15 - 57095-15-38RE-Question-Digital.docx Evaluating a...Ch. 15 - 57095-15-39RE-Question-Digital.docx Work Find the...Ch. 15 - 57095-15-40RE-Question-Digital.docx Work A 20-ton...Ch. 15 - 57095-15-41RE-Question-Digital.docx Using the...Ch. 15 - 57095-15-42RE-Question-Digital.docx Using the...Ch. 15 - 43. Evaluating a Line Integral Evaluate the line...Ch. 15 - 57095-15-44RE-Question-Digital.docx Area and...Ch. 15 - 57095-15-45RE-Question-Digital.docx Evaluating a...Ch. 15 - Evaluating a Line Integral In Exercises 45–50, use...Ch. 15 - 57095-15-47RE-Question-Digital.docx Evaluating a...Ch. 15 - 57095-15-48RE-Question-Digital.docx Evaluating a...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Graphing a Parametric SurfaceIn Exercise 65 and...Ch. 15 - Graphing a Parametric SurfaceIn Exercise 65 and...Ch. 15 - 57095-15-53RE-Question-Digital.docx Investigation...Ch. 15 - 57095-15-54RE-Question-Digital.docx Evaluating a...Ch. 15 - 57095-15-55RE-Question-Digital.docx Approximating...Ch. 15 - 57095-15-56RE-Question-Digital.docx Mass A...Ch. 15 - Verifying the Divergence Theorem In Exercises 57...Ch. 15 - Verifying the Divergence Theorem In Exercises 57...Ch. 15 - 57095-15-59RE-Question-Digital.docx Verifying...Ch. 15 - 57095-15-60RE-Question-Digital.docx Verifying...Ch. 15 - 57095-15-61RE-Question-Digital.docx Proof Prove...Ch. 15 - Heat Flux Consider a single heat source located at...Ch. 15 - Heat Flux Consider a single heat source located at...Ch. 15 - Moments of Inertia Consider a wire of density...Ch. 15 - Moments of Inertia Using the formulas from...Ch. 15 - Laplace's Equation Let F(x,y,z)=xi+yj+zk, and let...Ch. 15 - Greens Theorem Consider the line integral...Ch. 15 - Area Use a line integral to find the area bounded...Ch. 15 - Area Use a line integral to find the area bounded...Ch. 15 - Work The force field F(x,y)=(x+y)i+(x2+1)j acts on...Ch. 15 - Work The force field F(x,y)=3x2y2i+2x3yj is shown...Ch. 15 - 57095-15-11PS-Question-Digital.docx Proof Let S be...Ch. 15 - Area and Work How does the area of the ellipse...Ch. 15 - Verifying Identities (a) Let f and g be scalar...

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