Chapter 12: Q. 2 (page 988)

Fill in the blanks to complete each of the following theorem statements:
Let $f (x, y)$ be a function defined on an open subset $S$ of $ℝ^{2}$ . If the second-order partial derivatives of a function $f (x, y)$ are _ in a (an) _ subset $S$ of $ℝ^{2}$ , then $f_{x y} (x, y) = f_{y x} (x, y)$ at every point in $S$ .

Short Answer

Expert verified

Let $f (x, y)$ be a function defined on an open subset $S$ of $ℝ^{2}$ . If the second-order partial derivatives of a function $f (x, y)$ are continuous everywhere in an open subset $S$ of $ℝ^{2}$ , then $f_{x y} (x, y) = f_{y x} (x, y)$ at every point in $S$ .

Step by step solution

Step 1. Given information

Let $f (x, y)$ be a function defined on an open subset $S$ of $ℝ^{2}$ . If the second-order partial derivatives of a function $f (x, y)$ are _____ in a (an) _____ subset $S$ of $ℝ^{2}$ , then $f_{x y} (x, y) = f_{y x} (x, y)$ at every point in $S$ .

Step 2. Filling in the blanks

This is clairaut's theorem. The equality of mixed second-order partial derivatives can be seen on the open subset $S$ of $ℝ^{2}$ and the second order partial derivatives of a function should be continuous everywhere for the equality to hold good.

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Fill in the blanks to complete each of the following theorem statements:Let f(x,y)be a function defined on an open subset Sof ℝ2. If the second-order partial derivatives of a function f(x,y)are _____ in a (an) _____ subset Sof ℝ2, then fxy(x,y)=fyx(x,y)at every point in S.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Filling in the blanks

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Decision Maths

Probability and Statistics

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.