17. Describe the surface that has the following parametrization, and com- pute the Jacobian associated to the transformation: T(u, v) = cos u sin v sin u sin v COS U 0≤u≤ 2π, 0≤v≤T.
17. Describe the surface that has the following parametrization, and com- pute the Jacobian associated to the transformation: T(u, v) = cos u sin v sin u sin v COS U 0≤u≤ 2π, 0≤v≤T.
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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![**17. Describe the surface that has the following parametrization, and compute the Jacobian associated with the transformation:**
\[ T(u, v) = \begin{pmatrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{pmatrix}, \quad 0 \leq u \leq 2\pi, \; 0 \leq v \leq \pi. \]
**Explanation:**
This problem involves describing the geometry of a surface represented by a specific parametrization and computing the Jacobian matrix for this transformation. The transformation \( T(u, v) \) maps a point in the parameter space to a point on the surface.
1. **Describing the Surface:**
- The given transformation corresponds to the standard parametrization of a sphere. The parameters \( u \) and \( v \) represent angles in spherical coordinates.
- The \( x \)-coordinate is given by \( \cos u \sin v \),
- the \( y \)-coordinate by \( \sin u \sin v \),
- and the \( z \)-coordinate by \( \cos v \).
- The angle \( u \) ranges from \( 0 \) to \( 2\pi \), covering a full circle, while \( v \) ranges from \( 0 \) to \( \pi \), covering the top and bottom halves of the sphere.
2. **Computing the Jacobian:**
- The Jacobian matrix of the transformation is found by computing the partial derivatives of each coordinate with respect to the parameters \( u \) and \( v \). This matrix is crucial for understanding how area and volume transform under the parametrization.
This exploration of parametrization and Jacobian computation provides a deeper understanding of how mathematical transformations are used to describe geometric surfaces in space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc158a850-76a9-4504-97b9-8593e0926539%2Feddaac0b-2af5-4bbd-aae2-bc065a8f91f1%2F4xc12vn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**17. Describe the surface that has the following parametrization, and compute the Jacobian associated with the transformation:**
\[ T(u, v) = \begin{pmatrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{pmatrix}, \quad 0 \leq u \leq 2\pi, \; 0 \leq v \leq \pi. \]
**Explanation:**
This problem involves describing the geometry of a surface represented by a specific parametrization and computing the Jacobian matrix for this transformation. The transformation \( T(u, v) \) maps a point in the parameter space to a point on the surface.
1. **Describing the Surface:**
- The given transformation corresponds to the standard parametrization of a sphere. The parameters \( u \) and \( v \) represent angles in spherical coordinates.
- The \( x \)-coordinate is given by \( \cos u \sin v \),
- the \( y \)-coordinate by \( \sin u \sin v \),
- and the \( z \)-coordinate by \( \cos v \).
- The angle \( u \) ranges from \( 0 \) to \( 2\pi \), covering a full circle, while \( v \) ranges from \( 0 \) to \( \pi \), covering the top and bottom halves of the sphere.
2. **Computing the Jacobian:**
- The Jacobian matrix of the transformation is found by computing the partial derivatives of each coordinate with respect to the parameters \( u \) and \( v \). This matrix is crucial for understanding how area and volume transform under the parametrization.
This exploration of parametrization and Jacobian computation provides a deeper understanding of how mathematical transformations are used to describe geometric surfaces in space.
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