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Chapter 14: Problem 47

Evaluate the Jacobian for the transformation from spherical to rectangular coordinates: \(x=\rho \sin \varphi \cos \theta, y=\rho \sin \varphi \sin \theta, z=\rho \cos \varphi .\) Show that \(J(\rho, \varphi, \theta)=\rho^{2} \sin \varphi\)

Short Answer

Expert verified
Answer: The Jacobian determinant is $J(\rho, \varphi, \theta) = \rho^2 \sin \varphi$.

Step by step solution

01

Calculate the partial derivatives

First, we need to find the partial derivatives of \(x, y, z\) with respect to \(\rho, \varphi, \theta\). This will give us the elements of the Jacobian matrix. - \(\frac{\partial x}{\partial \rho} = \sin \varphi \cos \theta\) - \(\frac{\partial x}{\partial \varphi} = \rho \cos \varphi \cos \theta\) - \(\frac{\partial x}{\partial \theta} = -\rho \sin \varphi \sin \theta\) - \(\frac{\partial y}{\partial \rho} = \sin \varphi \sin \theta\) - \(\frac{\partial y}{\partial \varphi} = \rho \cos \varphi \sin \theta\) - \(\frac{\partial y}{\partial \theta} = \rho \sin \varphi \cos \theta\) - \(\frac{\partial z}{\partial \rho} = \cos \varphi\) - \(\frac{\partial z}{\partial \varphi} = -\rho \sin \varphi\) - \(\frac{\partial z}{\partial \theta} = 0\)
02

Construct the Jacobian matrix and find its determinant

Now that we have all the partial derivatives, we can construct the Jacobian matrix: $$ \begin{pmatrix} \frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \varphi} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \varphi} & \frac{\partial y}{\partial \theta} \\ \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \varphi} & \frac{\partial z}{\partial \theta} \end{pmatrix} = \begin{pmatrix} \sin \varphi \cos \theta & \rho \cos \varphi \cos \theta & -\rho \sin \varphi \sin \theta \\ \sin \varphi \sin \theta & \rho \cos \varphi \sin \theta & \rho \sin \varphi \cos \theta \\ \cos \varphi & -\rho \sin \varphi & 0 \end{pmatrix} $$ Now we must find the determinant of this matrix to obtain the Jacobian determinant \(J(\rho, \varphi, \theta)\). Using the definition of a determinant, we get: $$ \begin{aligned} J(\rho, \varphi, \theta) &= (\sin \varphi \cos \theta) \begin{vmatrix} \rho \cos \varphi \sin \theta & \rho \sin \varphi \cos \theta \\ -\rho \sin \varphi & 0 \end{vmatrix} - (\rho \cos \varphi \cos \theta) \begin{vmatrix} \sin \varphi \sin \theta & \rho \sin \varphi \cos \theta \\ \cos \varphi & 0 \end{vmatrix} + (-\rho \sin \varphi \sin \theta) \begin{vmatrix} \sin \varphi \sin \theta & \rho \cos \varphi \sin \theta \\ \cos \varphi & -\rho \sin \varphi \end{vmatrix} \\ &= (\sin \varphi \cos \theta) [(0)-(-\rho \sin \varphi)(\rho \sin \varphi \cos \theta)] - (\rho \cos \varphi \cos \theta) [(0)-(\cos \varphi)(\rho \sin \varphi \cos \theta)]. \end{aligned} $$ Simplifying, we get: $$ J(\rho, \varphi, \theta) = (\sin \varphi \cos \theta) [\rho^2 \sin^2 \varphi \cos \theta] $$ Further simplifying, we obtain the final answer: $$ J(\rho, \varphi, \theta) = \rho^2 \sin \varphi $$
03

Conclusion

We have shown that the Jacobian determinant for the transformation from spherical to rectangular coordinates is \(J(\rho, \varphi, \theta) = \rho^2 \sin \varphi\).

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Key Concepts

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

Understanding Spherical Coordinates
Spherical coordinates are a three-dimensional way of defining points in space. They are especially useful in scenarios involving spheres or radial systems, such as astronomy or electromagnetics. Instead of using linear measures, spherical coordinates use:
  • \(\rho\) (rho): The radial distance from the origin to the point.
  • \(\varphi\) (phi): The polar angle measured from the positive z-axis, similar to latitude.
  • \(\theta\) (theta): The azimuthal angle in the xy-plane from the positive x-axis, akin to longitude.
This system is helpful because many natural systems have spherical symmetry, meaning calculations using spherical coordinates can be simpler than using traditional Cartesian coordinates.
For transformations, understanding these angles and how they relate to their rectangular equivalents is crucial.
Converting to Rectangular Coordinates
Rectangular, or Cartesian coordinates, describe a point in space using x, y, and z values, representing horizontal, vertical, and depth dimensions respectively. Understanding how to convert between spherical and rectangular coordinates is essential in many fields.
The transformation equations are:
  • x = \(\rho \sin \varphi \cos \theta\): This equation defines the x-coordinate using the spherical coordinates.
  • y = \(\rho \sin \varphi \sin \theta\): Similarly, this describes the y-coordinate.
  • z = \(\rho \cos \varphi\): This equation assigns the z-coordinate.
These equations help transition calculations from a spherical to a rectangular system. By mastering these transformations, one can effectively work across different coordinate systems.
The Role of Partial Derivatives
Partial derivatives are a fundamental concept in calculus, representing the rate of change of a function concerning one variable, keeping the others constant. They are vital when dealing with multi-variable functions, like those in physics and engineering.In this context, partial derivatives help form the Jacobian matrix.
Each element in the Jacobian matrix is a partial derivative of the transformation equations, giving insight into how each coordinate relationship changes:
  • The derivative \(\frac{\partial x}{\partial \rho}\), for example, shows how x changes with respect to \(\rho\).
  • This process repeats for \(\varphi\) and \(\theta\) too, constructing a complete picture with the Jacobian matrix.
The determinant of this matrix, known as the Jacobian determinant, helps evaluate how volume or area transforms under spherical to rectangular conversion. It's especially important in physics and engineering where transformation behavior matters.

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Most popular questions from this chapter

To evaluate the following integrals carry out these steps. a. Sketch the original region of integration \(R\) in the xy-plane and the new region \(S\) in the uv-plane using the given change of variables. b. Find the limits of integration for the new integral with respect to \(u\) and \(v\) c. Compute the Jacobian. d. Change variables and evaluate the new integral. \(\iint_{R} x^{2} y d A,\) where \(R=\\{(x, y): 0 \leq x \leq 2, x \leq y \leq x+4\\}\) use \(x=2 u, y=4 v+2 u\)

An important integral in statistics associated with the normal distribution is \(I=\int_{-\infty}^{\infty} e^{-x^{2}} d x .\) It is evaluated in the following steps. a. Assume that \(I^{2}=\left(\int_{-\infty}^{\infty} e^{-x^{2}} d x\right)\left(\int_{-\infty}^{\infty} e^{-y^{2}} d y\right)=\) \(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^{2}-y^{2}} d x d y,\) where we have chosen the variables of integration to be \(x\) and \(y\) and then written the product as an iterated integral. Evaluate this integral in polar coordinates and show that \(I=\sqrt{\pi}\) b. Evaluate \(\int_{0}^{\infty} e^{-x^{2}} d x, \int_{0}^{\infty} x e^{-x^{2}} d x,\) and \(\int_{0}^{\infty} x^{2} e^{-x^{2}} d x\) (using part (a) if needed).

Evaluate the Jacobian for the transformation from cylindrical coordinates \((r, \theta, Z)\) to rectangular coordinates \((x, y, z): x=r \cos \theta, y=r \sin \theta, z=Z .\) Show that \(J(r, \theta, Z)=r\)

Compute the volume of the following solids. Tetrahedron A tetrahedron with vertices \((0,0,0),(a, 0,0)\) \((b, c, 0),\) and \((0,0, d),\) where \(a, b, c,\) and \(d\) are positive real numbers

The Jacobian is a magnification (or reduction) factor that relates the area of a small region near the point \((u, v)\) to the area of the image of that region near the point \((x, y)\) a. Suppose \(S\) is a rectangle in the \(u v\) -plane with vertices \(O(0,0)\) \(P(\Delta u, 0),(\Delta u, \Delta v),\) and \(Q(0, \Delta v)\) (see figure). The image of \(S\) under the transformation \(x=g(u, v), y=h(u, v)\) is a region \(R\) in the \(x y\) -plane. Let \(O^{\prime}, P^{\prime},\) and \(Q^{\prime}\) be the images of O, P, and \(Q,\) respectively, in the \(x y\) -plane, where \(O^{\prime}\) \(P^{\prime},\) and \(Q^{\prime}\) do not all lie on the same line. Explain why the coordinates of \(\boldsymbol{O}^{\prime}, \boldsymbol{P}^{\prime}\), and \(Q^{\prime}\) are \((g(0,0), h(0,0))\) \((g(\Delta u, 0), h(\Delta u, 0)),\) and \((g(0, \Delta v), h(0, \Delta v)),\) respectively. b. Use a Taylor series in both variables to show that $$\begin{aligned} &g(\Delta u, 0) \approx g(0,0)+g_{u}(0,0) \Delta u\\\ &g(0, \Delta v) \approx g(0,0)+g_{v}(0,0) \Delta v\\\ &\begin{array}{l} h(\Delta u, 0) \approx h(0,0)+h_{u}(0,0) \Delta u \\ h(0, \Delta v) \approx h(0,0)+h_{v}(0,0) \Delta v \end{array} \end{aligned}$$ where \(g_{u}(0,0)\) is \(\frac{\partial x}{\partial u}\) evaluated at \((0,0),\) with similar meanings for \(g_{v}, h_{u}\) and \(h_{v}\) c. Consider the vectors \(\overrightarrow{O^{\prime} P^{\prime}}\) and \(\overrightarrow{O^{\prime} Q^{\prime}}\) and the parallelogram, two of whose sides are \(\overrightarrow{O^{\prime} P^{\prime}}\) and \(\overrightarrow{O^{\prime} Q^{\prime}}\). Use the cross product to show that the area of the parallelogram is approximately \(|J(u, v)| \Delta u \Delta v\) d. Explain why the ratio of the area of \(R\) to the area of \(S\) is approximately \(|J(u, v)|\)
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