Chapter 12: Q. 58. (page 945)

For the partial derivatives given in Exercises 55–58, find the
most general form for a function of three variables, $f (x, y, z)$ ,
with the given partial derivative.
$\frac{\partial^{2} f}{\partial y \partial x} = 0$

Short Answer

Expert verified

The most general form of $f$ so that $\frac{\partial^{2} f}{\partial y \partial x} = 0$ is $f (x, y, z) = x h_{1} (z) + z h_{2} (y)$

Step by step solution

Given information

Given derivative is $\frac{\partial^{2} f}{\partial y \partial x} = 0$

The objective is to find the most general form of a function f(x, y, z) so that ∂2f∂y∂x=0

Suppose, $f (x, y, z) = x h_{1} (z) + z h_{2} (y)$

Then,

$\frac{d f}{d x} = h_{1} (z) + 0 \frac{d^{2} f}{d y d x} = 0$

Hence, the most general form of $f$ so that $\frac{\partial^{2} f}{\partial y \partial x} = 0$ is $f (x, y, z) = x h_{1} (z) + z h_{2} (y)$

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For the partial derivatives given in Exercises 55–58, find the
most general form for a function of three variables, $f (x, y, z)$ ,
with the given partial derivative.
$\frac{\partial^{2} f}{\partial y \partial x} = 0$

Short Answer

Step by step solution

Given information

The objective is to find the most general form of a function f(x, y, z) so that ∂2f∂y∂x=0

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For the partial derivatives given in Exercises 55–58, find themost general form for a function of three variables, f(x,y,z),with the given partial derivative.∂2f∂y∂x=0

Short Answer

Step by step solution

Given information

The objective is to find the most general form of a function f(x, y, z) so that ∂2f∂y∂x=0

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Mechanics Maths

Applied Mathematics

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

For the partial derivatives given in Exercises 55–58, find the
most general form for a function of three variables, $f (x, y, z)$ ,
with the given partial derivative.
$\frac{\partial^{2} f}{\partial y \partial x} = 0$