Q. 20 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Given

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

Step 2: To find

We have to find the relationship between f and g. 

Step 3: 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 y is independent of x.

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