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Chapter 14: Problem 22

Find the Jacobian \(\partial(x, y, z) / \partial(u, v, w)\) of the transformation a. \(x=u \cos v, \quad y=u \sin v, \quad z=w\) b. \(x=2 u-1, \quad y=3 v-4, \quad z=(1 / 2)(w-4)\)

Short Answer

Expert verified
Jacobian for (a) is 0; Jacobian for (b) is 3.

Step by step solution

01

Identify Function Components

In transformation (a), the functions are defined as: \(x = u \cos v\), \(y = u \sin v\), \(z = w\). In transformation (b), the functions are \(x = 2u - 1\), \(y = 3v - 4\), \(z = \frac{1}{2}(w - 4)\). The task is to find the Jacobian of each transformation.
02

Derivative Calculations for Transformation (a)

For transformation (a), calculate partial derivatives: \(\frac{\partial x}{\partial u} = \cos v\), \(\frac{\partial x}{\partial v} = -u \sin v\), \(\frac{\partial x}{\partial w} = 0\), \(\frac{\partial y}{\partial u} = \sin v\), \(\frac{\partial y}{\partial v} = u \cos v\), \(\frac{\partial y}{\partial w} = 0\), \(\frac{\partial z}{\partial u} = 0\), \(\frac{\partial z}{\partial v} = 0\), \(\frac{\partial z}{\partial w} = 1\).
03

Construct Jacobian Matrix for Transformation (a)

Combine the partial derivatives into a matrix: \[\begin{bmatrix}\cos v & -u \sin v & 0 \\sin v & u \cos v & 0 \0 & 0 & 1\end{bmatrix}\]
04

Compute Determinant of Jacobian for Transformation (a)

Find the determinant of the matrix: The determinant is \(\cos v \, (u \cos v \, \cdot 1) - (-u \sin v \, \cdot \sin v)\), which simplifies to \(0\). {
05

Derivative Calculations for Transformation (b)

For transformation (b), calculate partial derivatives: \(\frac{\partial x}{\partial u} = 2\), \(\frac{\partial x}{\partial v} = 0\), \(\frac{\partial x}{\partial w} = 0\), \(\frac{\partial y}{\partial u} = 0\), \(\frac{\partial y}{\partial v} = 3\), \(\frac{\partial y}{\partial w} = 0\), \(\frac{\partial z}{\partial u} = 0\), \(\frac{\partial z}{\partial v} = 0\), \(\frac{\partial z}{\partial w} = \frac{1}{2}\).
06

Construct Jacobian Matrix for Transformation (b)

Combine the partial derivatives into a matrix: \[\begin{bmatrix}2 & 0 & 0 \0 & 3 & 0 \0 & 0 & \frac{1}{2}\end{bmatrix}\]
07

Compute Determinant of Jacobian for Transformation (b)

Find the determinant of the matrix: The determinant is \(2 \cdot 3 \cdot \frac{1}{2} = 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transformation
In mathematics, a transformation refers to changing some variables into others using specific rules. This concept helps us understand how a system behaves when its coordinates are altered.
For instance, in the given transformations, we're mapping old variables \(u, v, w\) into new variables \(x, y, z\). This is accomplished by applying equations that define each new variable in terms of the old ones.
  • For transformation (a):
    • We have \(x = u \cos v\).
    • Similarly, \(y = u \sin v\).
    • Finally, \(z=w\) stays unchanged.
  • In transformation (b):
    • Variables transform as \(x=2u-1\).
    • Next, \(y=3v-4\).
    • Lastly, the equation \(z=\frac{1}{2}(w-4)\) transforms \(z\).
Transformations like these often simplify or change the problem's geometry, making other calculations, such as those involving Jacobians, more manageable.
Partial Derivatives
Partial derivatives are a fundamental tool in calculus when dealing with functions with more than one variable. They measure how a function changes as one variable changes, keeping others constant.
This concept was applied in solving the original exercise, where each component of the transformation had its partial derivatives calculated.
For transformation (a):
  • The partial derivative of \(x\) with respect to \(u\) is \(\cos v\).
  • With respect to \(v\), it becomes \(-u \sin v\).
  • The derivative with respect to \(w\) is 0 as \(x\) doesn't depend on \(w\).
For transformation (b):
  • Each partial derivative reflects how changing \(u, v, w\) affects their respective transformations like \(\frac{\partial x}{\partial u} = 2\).
  • Here \(\frac{\partial z}{\partial w} = \frac{1}{2}\) shows how the transformation affects \(z\) when \(w\) changes.
Partial derivatives are stacked together to build the Jacobian matrix, which we use to analyze transformations further.
Determinant of a Matrix
The determinant of a matrix is a unique number that helps us understand certain properties of linear transformations, notably their scaling effect on volume.
For our Jacobian matrices, the determinant tells us how the transformation scales space locally.
For transformation (a):
  • The Jacobian determinant was calculated to be 0, indicating that the transformation collapses space in some direction.
For transformation (b):
  • The determinant \(3\) implies that volumes are stretched by a factor of 3 under this transformation.
The determinant thus offers crucial insights into the behavior of transformations, highlighting whether they preserve, shrink, or expand the geometrical dimensions of the original space.

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Most popular questions from this chapter

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section \(8.7 .\) Evaluate the improper integrals as iterated integrals. $$\int_{1}^{\infty} \int_{e^{-}}^{1} \frac{1}{x^{3} y} d y d x$$

Let \(P_{0}\) be a point inside a circle of radius \(a\) and let \(h\) denote the distance from \(P_{0}\) to the center of the circle. Let \(d\) denote the distance from an arbitrary point \(P\) to \(P_{0} .\) Find the average value of \(d^{2}\) over the region enclosed by the circle. (Hint: Simplify your work by placing the center of the circle at the origin and \(P_{0}\) on the \(x\)-axis.)

Noncircular cylinder \(\quad\) A solid right (noncircular) cylinder has its base \(R\) in the \(x y\) -plane and is bounded above by the paraboloid \(z=x^{2}+y^{2} .\) The cylinder's volume is $$V=\int_{0}^{1} \int_{0}^{y}\left(x^{2}+y^{2}\right) d x d y+\int_{1}^{2} \int_{0}^{2-y}\left(x^{2}+y^{2}\right) d x d y$$ Sketch the base region \(R\) and express the cylinder's volume as a single iterated integral with the order of integration reversed. Then evaluate the integral to find the volume.

Use a CAS to change the Cartesian integrals into an equivalent polar integral and evaluate the polar integral. Perform the following steps in each exercise. a. Plot the Cartesian region of integration in the \(x y\)-plane. b. Change each boundary curve of the Cartesian region in part (a) to its polar representation by solving its Cartesian equation for \(r\) and \(\theta\) c. Using the results in part (b), plot the polar region of integration in the \(r \theta\)-plane. d. Change the integrand from Cartesian to polar coordinates. Determine the limits of integration from your plot in part (c) and evaluate the polar integral using the CAS integration utility. $$\int_{0}^{1} \int_{0}^{x / 2} \frac{x}{x^{2}+y^{2}} d y d x$$

Show that the centroid of the solid semiellipsoid of revolution \(\left(r^{2} / a^{2}\right)+\left(z^{2} / h^{2}\right) \leq 1, z \geq 0\) lies on the \(z\) -axis three-eighths of the way from the base to the top. The special case \(h=a\) gives a solid hemisphere. Thus, the centroid of a solid hemisphere lies on the axis of symmetry three eighths of the way from the base to the top.
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