Chapter 12: Q. 22 (page 944)

Let f(x, y, z) and g(x, y, z) be functions of two variables with the property that $\frac{\partial f}{\partial y} = \frac{\partial g}{\partial y}$ for every point $(x, y, z) \in R^{3}$ . What is the relationship between f and g?

Short Answer

Expert verified

The relationship is: $g (x, y, z) = f (x, y, z) + k (x, z)$ .

Step by step solution

Given

Given that: $\frac{\partial f}{\partial y} = \frac{\partial g}{\partial y}$ .

To find

We have to find the relationship between f and g.

Calculation

If $\frac{\partial f}{\partial y} = \frac{\partial g}{\partial y}$ then we can say that only y part of the functions is the same in both functions. And x, z are independent of y.

We can write them as: $g (x, y, z) = f (x, y, z) + k (x, z)$ , where $k (x, z)$ is a function of x and z.

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Let f(x, y, z) and g(x, y, z) be functions of two variables with the property that $\frac{\partial f}{\partial y} = \frac{\partial g}{\partial y}$ for every point $(x, y, z) \in R^{3}$ . What is the relationship between f and g?

Short Answer

Step by step solution

Given

To find

Calculation

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Let f(x, y, z) and g(x, y, z) be functions of two variables with the property that∂f∂y=∂g∂yfor every point x,y,z∈R3. What is the relationship between f and g?

Short Answer

Step by step solution

Given

To find

Calculation

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Most popular questions from this chapter

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Let f(x, y, z) and g(x, y, z) be functions of two variables with the property that $\frac{\partial f}{\partial y} = \frac{\partial g}{\partial y}$ for every point $(x, y, z) \in R^{3}$ . What is the relationship between f and g?