Let S be the portion of the cone with equation z² = x² + y² which lies between the planes z = 1 and z = 2, oriented so that the normal vectors point inward and upward. (a) Give S a parameterization of the form r(u, v), where (u, v) [a, b] × [c, d], making sure to be consistent with the orientation of S we want. (b) As you did in problem 2, use this parameterization to find the boundaries of S.
Let S be the portion of the cone with equation z² = x² + y² which lies between the planes z = 1 and z = 2, oriented so that the normal vectors point inward and upward. (a) Give S a parameterization of the form r(u, v), where (u, v) [a, b] × [c, d], making sure to be consistent with the orientation of S we want. (b) As you did in problem 2, use this parameterization to find the boundaries of S.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Certainly! Here is the transcription of the image text for an educational context:
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**3.** Let \( S \) be the portion of the cone with equation \( z^2 = x^2 + y^2 \) which lies between the planes \( z = 1 \) and \( z = 2 \), oriented so that the normal vectors point inward and upward.
**(a)** Give \( S \) a parameterization of the form \( \mathbf{r}(u, v) \), where \( (u, v) \in [a, b] \times [c, d] \), making sure to be consistent with the orientation of \( S \) we want.
**(b)** As you did in problem 2, use this parameterization to find the boundaries of \( S \).
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**(c)** Let \( \mathbf{F} \) be the vector field \( xy \, \mathbf{i} + z^2 \, \mathbf{j} - (x - z) \, \mathbf{k} \). Stokes' theorem relates the curl integral of \( \mathbf{F} \) over \( S \) to the circulation integral of \( \mathbf{F} \) along the boundary of \( S \). The surface in this problem has two boundaries which we can call \( C \) and \( C' \). This means that in this case, Stokes' theorem tells us that
\[
\iint_S \underline{\hspace{1cm}} \, dy \wedge dz + \underline{\hspace{1cm}} \, dz \wedge dx + \underline{\hspace{1cm}} \, dx \wedge dy = \int_C \underline{\hspace{1cm}} \, dx + \underline{\hspace{1cm}} \, dy + \underline{\hspace{1cm}} \, dz + \int_{C'} \underline{\hspace{1cm}} \, dx + \underline{\hspace{1cm}} \, dy + \underline{\hspace{1cm}} \, dz.
\]
Fill in the blanks, and describe \( C \) and \( C' \) geometrically. You do not have to evaluate the integrals.
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Note: For effective learning, it may be helpful to explore visual aids such as diagrams of the cone and the surfaces](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2F39f4c4fd-6d5b-43c9-956c-dc2f3c7d0f95%2F9nd66no_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcription of the image text for an educational context:
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**3.** Let \( S \) be the portion of the cone with equation \( z^2 = x^2 + y^2 \) which lies between the planes \( z = 1 \) and \( z = 2 \), oriented so that the normal vectors point inward and upward.
**(a)** Give \( S \) a parameterization of the form \( \mathbf{r}(u, v) \), where \( (u, v) \in [a, b] \times [c, d] \), making sure to be consistent with the orientation of \( S \) we want.
**(b)** As you did in problem 2, use this parameterization to find the boundaries of \( S \).
---
**(c)** Let \( \mathbf{F} \) be the vector field \( xy \, \mathbf{i} + z^2 \, \mathbf{j} - (x - z) \, \mathbf{k} \). Stokes' theorem relates the curl integral of \( \mathbf{F} \) over \( S \) to the circulation integral of \( \mathbf{F} \) along the boundary of \( S \). The surface in this problem has two boundaries which we can call \( C \) and \( C' \). This means that in this case, Stokes' theorem tells us that
\[
\iint_S \underline{\hspace{1cm}} \, dy \wedge dz + \underline{\hspace{1cm}} \, dz \wedge dx + \underline{\hspace{1cm}} \, dx \wedge dy = \int_C \underline{\hspace{1cm}} \, dx + \underline{\hspace{1cm}} \, dy + \underline{\hspace{1cm}} \, dz + \int_{C'} \underline{\hspace{1cm}} \, dx + \underline{\hspace{1cm}} \, dy + \underline{\hspace{1cm}} \, dz.
\]
Fill in the blanks, and describe \( C \) and \( C' \) geometrically. You do not have to evaluate the integrals.
---
Note: For effective learning, it may be helpful to explore visual aids such as diagrams of the cone and the surfaces
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