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Chapter 14: Problem 21

Compute the Jacobian \(J(u, v)\) for the following transformations. $$T: x=(u+v) / \sqrt{2}, y=(u-v) / \sqrt{2}$$

Short Answer

Expert verified
Answer: The Jacobian matrix for the given transformation is: $$J(u, v) = \begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} \\ \dfrac{1}{\sqrt{2}} & -\dfrac{1}{\sqrt{2}} \end{bmatrix}$$

Step by step solution

01

Compute Partial Derivatives

We will calculate the partial derivatives of x and y with respect to u and v: 1. \(\dfrac{\partial x}{\partial u} = \dfrac{\partial}{\partial u} \bigg(\dfrac{u+v}{\sqrt{2}}\bigg)\) 2. \(\dfrac{\partial x}{\partial v} = \dfrac{\partial}{\partial v}\bigg(\dfrac{u+v}{\sqrt{2}}\bigg)\) 3. \(\dfrac{\partial y}{\partial u} = \dfrac{\partial }{\partial u}\bigg(\dfrac{u-v}{\sqrt{2}}\bigg)\) 4. \(\dfrac{\partial y}{\partial v} = \dfrac{\partial}{\partial v}\bigg(\dfrac{u-v}{\sqrt{2}}\bigg)\)
02

Evaluate Partial Derivatives

Apply the differentiation and simplify the results: 1. \(\dfrac{\partial x}{\partial u} = \dfrac{1}{\sqrt{2}}\) 2. \(\dfrac{\partial x}{\partial v} = \dfrac{1}{\sqrt{2}}\) 3. \(\dfrac{\partial y}{\partial u} = \dfrac{1}{\sqrt{2}}\) 4. \(\dfrac{\partial y}{\partial v} = -\dfrac{1}{\sqrt{2}}\)
03

Construct the Jacobian Matrix

Now, arrange the partial derivatives into a 2x2 matrix: $$J(u,v) = \begin{bmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} \\ \dfrac{1}{\sqrt{2}} & -\dfrac{1}{\sqrt{2}} \end{bmatrix}$$ Hence, the Jacobian for the given transformation T is: $$J(u, v) = \begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} \\ \dfrac{1}{\sqrt{2}} & -\dfrac{1}{\sqrt{2}} \end{bmatrix}$$

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Key Concepts

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

Partial Derivatives
Partial derivatives are a fundamental concept when dealing with multivariable functions. They represent the rate of change of a function with respect to one of its variables while keeping the other variables constant.
For example, if you have a function like \(z = f(x, y)\), the partial derivative \(\frac{\partial z}{\partial x}\) shows how \(z\) changes as \(x\) changes, assuming \(y\) remains the same.
  • To compute partial derivatives, differentiate the function concerning one variable while treating the other variables like constants.
  • In our exercise, we found the partial derivatives of \(x\) and \(y\) with respect to \(u\) and \(v\). This is crucial for forming the Jacobian matrix.
Partial derivatives lay the groundwork for optimizing functions and analyzing surfaces in higher dimensions.
Transformation
A transformation in mathematics refers to a function that maps points from one coordinate system to another. It's like changing the "view" of the data.
In this exercise, the transformation \(T\) is given by \(x=(u+v) / \sqrt{2}\) and \(y=(u-v) / \sqrt{2}\).
  • This transformation maps inputs \(u\) and \(v\) in a new system to outputs \(x\) and \(y\) in a different system.
  • Understanding transformations are essential, especially when solving complex real-world problems involving rotations, scaling, and translations.
Such transformations allow us to manipulate and understand multi-dimensional spaces efficiently, making them invaluable in physics and engineering.
Linear Algebra
Linear algebra is the study of vectors, vector spaces, and linear transformations. It provides the tools for dealing with multidimensional spaces efficiently.
In our scenario, the Jacobian matrix is a key concept from linear algebra. It describes how the function changes linearly around a point.
  • The Jacobian matrix is constructed using partial derivatives, indicating how each input affects each output.
  • Its determinant can provide information about the volume distortion during transformation.
Grasping these concepts enables deeper insights into systems of linear equations, which is vital for countless applications in science and technology.
Differentiation
Differentiation is a fundamental computational technique used in calculus to determine the rate of change of a function. It is the process of finding a derivative.
In multivariable functions, differentiation can be more complex as it involves calculating partial derivatives.
  • The rules of differentiation, such as the product rule and chain rule, still apply here but are adapted for multiple variables.
  • In this exercise, differentiation was used to compute the partial derivatives required to form the Jacobian matrix.
Mastering differentiation is vital for analyzing and interpreting the behavior of functions, especially functions of several variables.

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Most popular questions from this chapter

Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that \(a, b, c, r, R,\) and \(h\) are positive constants. Ellipsoid Find the volume of an ellipsoid with axes of length \(2 a\) \(2 b,\) and \(2 c\)

Use polar coordinates to find the centroid of the following constant-density plane regions. The semicircular disk \(R=\\{(r, \theta): 0 \leq r \leq 2,0 \leq \theta \leq \pi\\}\)

Area formula The area of a region enclosed by the polar curve \(r=g(\theta)\) and the rays \(\theta=\alpha\) and \(\theta=\beta,\) where \(\beta-\alpha \leq 2 \pi\) is \(A=\frac{1}{2} \int_{\alpha}^{\beta} r^{2} d \theta\). Prove this result using the area formula with double integrals.

Evaluate the Jacobians \(J(u, v, w)\) for the following transformations. $$x=v w, y=u w, z=u^{2}-v^{2}$$

Consider the region \(R\) bounded by three pairs of parallel planes: \(a x+b y=0, a x+b y=1, c x+d z=0\) \(c x+d z=1, e y+f z=0,\) and \(e y+f z=1,\) where \(a, b, c, d, e\) and \(f\) are real numbers. For the purposes of evaluating triple integrals, when do these six planes bound a finite region? Carry out the following steps. a. Find three vectors \(\mathbf{n}_{1}, \mathbf{n}_{2},\) and \(\mathbf{n}_{3}\) each of which is normal to one of the three pairs of planes. b. Show that the three normal vectors lie in a plane if their triple scalar product \(\mathbf{n}_{1} \cdot\left(\mathbf{n}_{2} \times \mathbf{n}_{3}\right)\) is zero. c. Show that the three normal vectors lie in a plane if ade \(+\) bcf \(=0\) d. Assuming \(\mathbf{n}_{1}, \mathbf{n}_{2},\) and \(\mathbf{n}_{3}\) lie in a plane \(P,\) find a vector \(\mathbf{N}\) that is normal to \(P\). Explain why a line in the direction of \(\mathbf{N}\) does not intersect any of the six planes, and thus the six planes do not form a bounded region. e. Consider the change of variables \(u=a x+b y, v=c x+d z\) \(w=e y+f z .\) Show that $$ J(x, y, z)=\frac{\partial(u, v, w)}{\partial(x, y, z)}=-a d e-b c f $$ What is the value of the Jacobian if \(R\) is unbounded?
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