Q. 22 - Calculus

Question

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 $\left(x, y, z\right)\in R^3$. What is the relationship between f and g?

Step-by-Step Solution

Answer

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

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 x, z are independent of y.

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