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Chapter 12: Problem 41

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\sqrt{x^{2}+y^{2}}$$

Short Answer

Expert verified
Answer: The function \(f(x, y) = \sqrt{x^{2}+y^{2}}\) is continuous at all points in \(\mathbb{R}^{2}\).

Step by step solution

01

Analyze the function

Observe the function \(f(x, y) = \sqrt{x^{2}+y^{2}}\). We can see that this function is non-negative for all values of \(x\) and \(y\), since the square root of a non-negative number is also non-negative. Also, the function is defined for all points in \(\mathbb{R}^{2}\).
02

Apply the definition of continuity

Recall that a function is continuous at a point \((a, b)\) if the limit of the function as \((x, y) \rightarrow (a, b)\) exists and is equal to the value of the function at that point, i.e. $$\lim_{(x, y) \rightarrow (a, b)}f(x, y) = f(a, b)$$. In the context of our function, this means that for any point \((a, b) \in \mathbb{R}^2\), the limit of \(f(x, y)\) as \((x, y)\rightarrow(a, b)\) should exist, and be equal to \(f(a, b)\).
03

Determine the limit of the function

In the case of our function \(f(x, y) = \sqrt{x^{2}+y^{2}}\), we know that \(x^2+y^2\) is non-negative for all \((x,y) \in \mathbb{R}^2\). Thus, the inner expression is always well-defined and continuous for all points in \(\mathbb{R}^2\). We also know that the square root function is continuous over its domain, i.e., the set of non-negative real numbers.
04

Conclude that the function is continuous

Since \(f(x, y) = \sqrt{x^{2}+y^{2}}\) is a composition of continuous functions, it must be continuous at all points of \(\mathbb{R}^{2}\). Therefore, the function \(f(x, y)\) is continuous for all points \((x, y) \in \mathbb{R}^2\).

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Most popular questions from this chapter

Flow in a cylinder Poiseuille's Law is a fundamental law of fluid dynamics that describes the flow velocity of a viscous incompressible fluid in a cylinder (it is used to model blood flow through veins and arteries). It says that in a cylinder of radius \(R\) and length \(L,\) the velocity of the fluid \(r \leq R\) units from the center-line of the cylinder is \(V=\frac{P}{4 L \nu}\left(R^{2}-r^{2}\right),\) where \(P\) is the difference in the pressure between the ends of the cylinder and \(\nu\) is the viscosity of the fluid (see figure). Assuming that \(P\) and \(\nu\) are constant, the velocity \(V\) along the center line of the cylinder \((r=0)\) is \(V=k R^{2} / L,\) where \(k\) is a constant that we will take to be \(k=1.\) a. Estimate the change in the centerline velocity \((r=0)\) if the radius of the flow cylinder increases from \(R=3 \mathrm{cm}\) to \(R=3.05 \mathrm{cm}\) and the length increases from \(L=50 \mathrm{cm}\) to \(L=50.5 \mathrm{cm}.\) b. Estimate the percent change in the centerline velocity if the radius of the flow cylinder \(R\) decreases by \(1 \%\) and the length \(L\) increases by \(2 \%.\) c. Complete the following sentence: If the radius of the cylinder increases by \(p \%,\) then the length of the cylinder must increase by approximately __________ \(\%\) in order for the velocity to remain constant.

Prove that for the plane described by \(f(x, y)=A x+B y,\) where \(A\) and \(B\) are nonzero constants, the gradient is constant (independent of \((x, y)\) ). Interpret this result.

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The pressure, temperature, and volume of an ideal gas are related by \(P V=k T,\) where \(k>0\) is a constant. Any two of the variables may be considered independent, which determines the third variable. a. Use implicit differentiation to compute the partial derivatives \(\frac{\partial P}{\partial V} \frac{\partial T}{\partial P},\) and \(\frac{\partial V}{\partial T}\) b. Show that \(\frac{\partial P}{\partial V} \frac{\partial T}{\partial P} \frac{\partial V}{\partial T}=-1 .\) (See Exercise 67 for a generalization.)

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