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Chapter 10: Problem 17

\(f(x, y)=y\)

Short Answer

Expert verified
The function \(f(x, y) = y\) directly gives the second coordinate (y-coordinate) as the output for any 2D input (x, y). The function does not affect or care about the value for x at all.

Step by step solution

01

Identifying the function form

The function is given as \(f(x, y)=y\), which is a very simple function that directly points to its second coordinate without any operations or transformations. The input is a 2D point and the output is the y-coordinate of that point.

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Key Concepts

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

2D point
In mathematics, a 2D point is essentially a pair of numbers that identify a position on a flat surface. This surface is often referred to as a plane. A 2D point is typically represented as
  • (x, y), where x is the horizontal component and y is the vertical component.

Each of these components is a coordinate that specifies how far the point is from the origin along its respective axis.
Imagine drawing a point on a piece of paper, where the intersection of two lines (axes) is the origin. The x value tells you how far to move left or right, and the y value tells you how far to move up or down. This simple coordinate pair allows the point to be precisely located within the 2D space.
coordinate system
A coordinate system is a method of using numbers to represent a location in space. For a 2D space, we commonly use the Cartesian coordinate system. This system comprises two perpendicular lines, known as axes, which intersect at a point called the origin.

  • The horizontal line is the x-axis.
  • The vertical line is the y-axis.
These axes divide the plane into four quadrants. Coordinates are given as ordered pairs
  • (x, y), where the first number corresponds to a position along the x-axis and the second number corresponds to a position along the y-axis.

In the problem, when we see f(x, y), (x, y) represents the coordinates of points in such a system. The function determines a rule or relationship between these two coordinates. This coordinate system helps not only in simplifying calculations but also in visualizing the geometric interpretations of algebraic equations.
function form
The function form in mathematics refers to the specific way a function is expressed. In the given exercise, the function is defined as f(x, y) = y. This is a straightforward function that outputs the y-coordinate of a 2D point directly.
  • This function demonstrates a relationship between the input, which is a 2D point, and the output, which is simply the y-coordinate of this point.

The function f(x, y) = y can be seen as a way of extracting information specifically about the vertical position of points in a coordinate system.
Its simplicity lies in its lack of operations on the x-coordinate, emphasizing a direct readout of the y-component. Understanding this basic form assists in recognizing and predicting patterns and behaviors in more complex functions that one might encounter within mathematics.

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Most popular questions from this chapter

\(f(x)=e^{-x}\) \(0<\) \(x<\) 1

$$ \frac{\partial^{2} u}{\partial t^{2}} $$ \(=\frac{\partial^{2} u}{\partial x^{2}}+x \sin t\), \(0\)<\(x\)<\(\pi\), \(t>0\) $$ u(0, t) $$ \(=u(\pi, t)=0\) \(t>0\), $$ u(x, 0)=0 $$, \(0\)<\(x\)<\(\pi\), $$ \frac{\partial u}{\partial t}(x, 0) $$ \(=0\), \(0\)<\(x\)<\(\pi\)

\(f(x)=x(\pi-x)\)

Fluid Flow Around a Corner. The stream lines that describe the fluid flow around a corner (see Figure 10.28\()\) are given by \(\phi(x, y)=k,\) where \(k\) is a constant and \(\phi\) the stream function, satisfies the boundary value problem (a) Using the results of Problem \(22,\) show that this problem can be reduced to finding the flow above a flat plate (see Figure 10.29 ). That is, show that the problem reduces to finding the solution to \(\frac{\partial^{2} \psi}{\partial u^{2}}+\frac{\partial^{2} \psi}{\partial v^{2}}=0, \quad v>0, \quad-\infty< u <\infty,\) \(\psi(u, 0)=0, \quad-\infty< u <\infty,\) where \(\quad \psi \quad\) and \(\quad \phi \quad\) are related as follows: \(\psi(u, v)=\phi(x(u, v), y(u, v)) \quad\) with the \(\quad\) mapping between \((u, v)\) and \((x, y)\) given by the analytic function \(f(z)=z^{2} .\) (b) Verify that a nonconstant solution to the problem in part (a) is given by \(\psi(u, v)=v.\) (c) Using the result of part (b), find a stream function \(\phi(x, y)\) for the original problem.

$$\frac{\partial u}{\partial t}=3 \frac{\partial^{2} u}{\partial x^{2}}+x, 0<\mathcal{x}<\pi,t>0,$$ $$u(0, t)=u(\pi, t)=0, \quad t>0,$$ $$u(x, 0)=\sin x,0<\mathcal{x}<\pi$$
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