Chapter 12: Q. 19 (page 944)

Let f(x, y) and g(x, y) be functions of two variables with the property that $\frac{\partial f}{\partial x} = \frac{\partial g}{\partial x}$ for every point $(x, y) \in R^{2}$ . What is the relationship between f and g?

Short Answer

Expert verified

The relationship is: $g (x, y) = f (x, y) + k (y)$ , where $k (y)$ is a function of y.

Step by step solution

Given

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

To find

We have to find the relationship between f and g.

Calculation

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

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

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Let f(x, y) and g(x, y) be functions of two variables with the property that $\frac{\partial f}{\partial x} = \frac{\partial g}{\partial x}$ for every point $(x, y) \in R^{2}$ . 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) and g(x, y) be functions of two variables with the property that ∂f∂x=∂g∂xfor every point x,y∈R2. 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) and g(x, y) be functions of two variables with the property that $\frac{\partial f}{\partial x} = \frac{\partial g}{\partial x}$ for every point $(x, y) \in R^{2}$ . What is the relationship between f and g?