Q. 19 - Calculus

Question

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 $\left(x,y\right)\in R^2$. What is the relationship between f and g?

Step-by-Step Solution

Answer

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

Step 1: Given

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

Step 2: To find

We have to find the relationship between f and g.

Step 3: 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: $g\left(x,y\right)=f\left(x,y\right)+k\left(y\right)$, where $k\left(y\right)$ is a function of y.