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Chapter 15: Problem 3

a. Solve the system \({u=3 x+2 y, v=x+4 y\) for \(x\) and \(y\) in terms of \(u\) and \(v\). Then find the value of the Jacobian \(\partial(x, y) / \partial(u, v)\). b. Find the image under the transformation \(u=3 x+2 y\) \(v=x+4 y\) of the triangular region in the \(x y-\)plane bounded by the \(x-\)axis, the \(y\)-axis, and the line \(x+y=1\). Sketch the transformed region in the \(u v\)-plane.

Short Answer

Expert verified
Solve using equations \(x = \frac{2u - 3v}{5}\), \(y = \frac{3v - u}{10}\). Jacobian is \(\frac{3}{50}\). Transform to get region with vertices \((0,0), (3,1), (2,4)\).

Step by step solution

01

Write equations from the given transformation

We have the transformation equations:\[u = 3x + 2y\] and \[v = x + 4y\]. The goal is to express \(x\) and \(y\) in terms of \(u\) and \(v\).
02

Solve for x

To find \(x\), we manipulate the equations:Multiply the second equation by 2: \(2v = 2x + 8y\).Subtract this from the first equation:\( u - 2v = 3x + 2y - 2x - 8y = x - 6y\)So, \(x = u - 2v + 6y\).
03

Solve for y and simplify

Substitute \(x = u - 2v + 6y\) into \(v = x + 4y\):\[ v = (u - 2v + 6y) + 4y = u - 2v + 10y \]This gives us \(10y = v + 2v - u\) or \(10y = 3v - u\).Thus, \(y = \frac{3v - u}{10}\).
04

Substitute y to find x

Substitute \(y = \frac{3v - u}{10}\) back into \(x = u - 2v + 6y\):\[ x = u - 2v + 6\left(\frac{3v - u}{10}\right) \]\[ x = u - 2v + \frac{18v - 6u}{10} \]Distribute and combine to get:\[ x = u - \frac{60v}{10} + \frac{18v}{10} - \frac{6u}{10} \]Simplify to:\[ x = \frac{4u - 6v}{10} \] or \[ x = \frac{2u - 3v}{5} \].
05

Find the Jacobian

The Jacobian \(\frac{\partial(x, y)}{\partial(u, v)}\) is computed as:\[J = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}\]From previously derived equations:\[ x = \frac{2u - 3v}{5} \] and \[ y = \frac{3v - u}{10} \].The partial derivatives are:\(\frac{\partial x}{\partial u} = \frac{2}{5}\), \(\frac{\partial x}{\partial v} = -\frac{3}{5}\), \(\frac{\partial y}{\partial u} = -\frac{1}{10}\), and \(\frac{\partial y}{\partial v} = \frac{3}{10}\).Evaluate the determinant:\[J = \frac{2}{5} \cdot \frac{3}{10} - \left(-\frac{3}{5} \cdot -\frac{1}{10}\right) = \frac{6}{50} - \frac{3}{50} = \frac{3}{50}\].
06

Rewrite region boundaries in uv-plane

The triangle in the \(xy\)-plane is defined by \((x = 0, y = 0)\), \((y = 0, x=1)\), and \((x+y=1)\).Using transformations for these:- At \((0,0)\) in \(xy\)-plane, \((u,v)= (0,0)\).- At \((1,0)\), \((u,v) = (3,1)\).- At \((0,1)\), \((u,v) = (2,4)\).Express line \(x + y = 1\) as \(u = v + 1\).
07

Sketch and describe the region transformation

The transformed region in the \(uv\)-plane forms another triangle bounded by the lines:- \(u=0\)- \(v=0\)- \(u = v + 1\)These three lines form a triangular region with vertices at: \((0,0), (3,1), (2,4)\). Plot these points and lines on a graph.

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Key Concepts

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

Systems of Equations
In mathematics, particularly in linear algebra, systems of equations are sets of equations with multiple variables that we aim to solve simultaneously. For example, in a two-variable case, we often have equations that define relationships between two variables, say \(x\) and \(y\). In our exercise, the transformation equations \(u = 3x + 2y\) and \(v = x + 4y\) form such a system.Solving a system means finding the values of variables that satisfy all equations at the same time. There are various methods to solve these, such as substitution, elimination, or using matrices. In our particular problem, substitution was used by first manipulating the equations to express \(x\) and \(y\) in terms of \(u\) and \(v\). The strategic use of substitution helps to simplify one variable in terms of the others, eventually leading to a solution for each variable independently. Understanding systems of equations is crucial as it provides a way to model and solve real-world problems where multiple conditions must be met simultaneously.
  • Definitions: Equations with multiple variables solved together.
  • Methods: Substitution and elimination.
  • Importance: Models complex real-world situations.
Transformation
A transformation in mathematics refers to an operation that changes the position or form of objects. In the context of our exercise, the transformation connects two different coordinate systems鈥擻(xy\) and \(uv\) planes.When we talk about a transformation from \(xy\) to \(uv\), it means that we change each point defined by coordinates \((x, y)\) to a new point \((u, v)\), using a set of transformation rules like \(u = 3x + 2y\) and \(v = x + 4y\). This process involves finding invertible functions, which allow switching between the two sets of coordinates, maintaining the relationship between objects.The significance of a transformation lies in its ability to simplify complex problems by changing the point of view. For example, finding the Jacobian in this transformation helps measure how the area in the \(xy\)-plane gets altered as it moves into the \(uv\)-plane.
  • Concept: Change of position/form of objects.
  • Purpose: Simplify problems by altering coordinates.
  • Usefulness: Offers different perspectives and simplifies calculations.
Triangular Region
In geometry and calculus, the concept of a triangular region plays an essential role in integrations and transformations. A triangular region is a polygon with three sides, forming a closed shape.In our exercise, we have a triangular region bounded by the lines, specifically the \(x\)-axis, \(y\)-axis, and a line \(x + y = 1\). Each of these lines forms a boundary of the triangle in the \(xy\)-plane. This region contains points that satisfy all conditions defined by these boundaries.Through the transformation equations, this region transforms into a different triangle in the \(uv\)-plane with distinct vertices \((0,0), (3,1), (2,4)\). Understanding this transformation is key, as it helps visualize how geometric shapes and boundaries shift between various coordinate systems. Sketching these regions with the correct boundaries and shapes is a crucial skill in mathematics, allowing for a visual interpretation of algebraic operations.
  • Definition: Polygon with three sides鈥攂asic closed shape.
  • Boundaries: Define limits of the region.
  • Visual Importance: Helps understand transformations and bounds in different planes.

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