Q. 21 - Calculus

Question

Let f(x, y) and g(x, y) be functions of two variables with the property that $\frac{\partial f}{\partial x}=\frac{\partial g}{\partial x}$and $\frac{\partial f}{\partial y}=\frac{\partial g}{\partial y}$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:$f\left(x,y\right)=g\left(x,y\right)$.

Step 1: Given

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

Step 2: To find

We have to find the relationship between f and g. 

Step 3: Calculation

If $\frac{\partial f}{\partial x}=\frac{\partial g}{\partial x} \text{and} \frac{\partial f}{\partial y}=\frac{\partial g}{\partial y}$ then we can say that both x and y parrs of the functions are equal.

We can write it as: $f\left(x,y\right)=g\left(x,y\right)$