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Chapter 12: Problem 52

Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). $$f(w, x, y, z)=\sqrt{1-w^{2}-x^{2}-y^{2}-z^{2}}$$

Short Answer

Expert verified
Answer: The domain of the function is all points (w, x, y, z) in 4-dimensional space such that \(w^2 + x^2 + y^2 + z^2 \leq 1\), which represents the points inside or on the surface of a sphere of radius 1 centered at the origin (0,0,0,0). The domain can be written as: $$\{(w,x,y,z) \in \mathbb{R}^4 : w^2 + x^2 + y^2 + z^2 \leq 1\}$$

Step by step solution

01

Write down the inequality for the real output

Since the output of a square root should be real, we need to have the expression under the square root to be greater than or equal to zero, as follows: $$1 - w^2 - x^2 - y^2 - z^2 \geq 0$$
02

Rearrange the inequality so that all terms are on the left side

We want to rearrange the inequality by moving all terms to the left side: $$w^2 + x^2 + y^2 + z^2 \leq 1$$
03

Interpret the inequality geometrically

Observe that the left-hand side of the inequality is the sum of the squares of the coordinates in 4-dimensional space. Therefore, geometrically, the inequality represents the equation of a sphere in 4D space, with radius 1 and centered at the origin (0,0,0,0).
04

Describe the domain

From the previous step, we see that the inequality \(w^2 + x^2 + y^2 + z^2 \leq 1\) represents a sphere in 4D space. Therefore, the domain of the function is all points inside or on the surface of this sphere, including the origin. This can be written as: $$\{(w, x, y, z) \in \mathbb{R}^4 : w^2 + x^2 + y^2 + z^2 \leq 1\}$$

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Most popular questions from this chapter

The equation \(x^{2 n}+y^{2 n}+z^{2 n}=1,\) where \(n\) is a positive integer, describes a flattened sphere. Define the extreme points to be the points on the flattened sphere with a maximum distance from the origin. a. Find all the extreme points on the flattened sphere with \(n=2\) What is the distance between the extreme points and the origin? b. Find all the extreme points on the flattened sphere for integers \(n>2 .\) What is the distance between the extreme points and the origin? c. Give the location of the extreme points in the limit as \(n \rightarrow \infty\). What is the limiting distance between the extreme points and the origin as \(n \rightarrow \infty ?\)

(Adapted from 1938 Putnam Exam) Consider the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1\) and the plane \(P\) given by \(A x+B y+C z+1=0\). Let \(h=\left(A^{2}+B^{2}+C^{2}\right)^{-1 / 2}\) and \(m=\left(a^{2} A^{2}+b^{2} B^{2}+c^{2} C^{2}\right)^{1 / 2}\) a. Find the equation of the plane tangent to the ellipsoid at the point \((p, q, r)\) b. Find the two points on the ellipsoid at which the tangent plane is parallel to \(P\) and find equations of the tangent planes. c. Show that the distance between the origin and the plane \(P\) is \(h\) d. Show that the distance between the origin and the tangent planes is \(h m\) e. Find a condition that guarantees that the plane \(P\) does not intersect the ellipsoid.

The flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1\). $$u(x, t)=e^{-t}(2 \sin x+3 \cos x)$$

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0.$$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=x\left(x^{2}-3 y^{2}\right)$$

Let \(f\) be a differentiable function of one or more variables that is positive on its domain. a. Show that \(d(\ln f)=\frac{d f}{f}.\) b. Use part (a) to explain the statement that the absolute change in \(\ln f\) is approximately equal to the relative change in \(f.\) c. Let \(f(x, y)=x y,\) note that \(\ln f=\ln x+\ln y,\) and show that relative changes add; that is, \(d f / f=d x / x+d y / y.\) d. Let \(f(x, y)=x / y,\) note that \(\ln f=\ln x-\ln y,\) and show that relative changes subtract; that is \(d f / f=d x / x-d y / y.\) e. Show that in a product of \(n\) numbers, \(f=x_{1} x_{2} \cdots x_{n},\) the relative change in \(f\) is approximately equal to the sum of the relative changes in the variables.
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