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Chapter 12: Q 69. (page 932)

Let S be a subset of $R^{2} o r R^{3}$ . Prove that $\partial (\partial S) \subseteq \partial S$

Short Answer

Expert verified

It is proved that ifS is the subset then $\partial (\partial S) \subseteq \partial S$

Step by step solution

01

Step 1. Given information.

We have givenS is a subset of $R^{2} o r R^{3}$ .

02

Prove the given statement.

It is already proved that $\partial S$ is a closed set.

Substitute $\partial S$ for A in the theorem which states that for every closed set A, $\partial A \subseteq A$ .

Thus, $\partial (\partial S) \subseteq \partial S$

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Most popular questions from this chapter

In Exercises $21 - 28$ , find the directional derivative of the given function at the specified point $P$ and in the direction of the given unit vector $u$ .

$f (x, y) = \sqrt{\frac{y}{x}}$ at $P = (4, 9), u = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$

Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.

$f (x, y) = \frac{x}{y^{'}} P = (4, 3)$

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Find \frac{\partial f}{\partial s} and \frac{\partial f}{\partial t} using the preceding results .$

Evaluate the following limits, or explain why the limit does not exist.

$\lim_{(x, y) \to (4, π)} \tan^{- 1} (\frac{y}{x})$

Describe the meanings of each of the following mathematical expressions :

$\frac{\partial w}{\partial y}$

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