June 1st, 2017, 02:23 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Integration
The functions $f(z)$ and $g(z)$ are such that the function $F= (2x+yf(z))i+(2y+xf(z))j+xyg(z)k$ can be written as a gradient of some scalar function. Pick up the possible choices for $f$ and $g$ from the following. A) $f(z) = z^3$ and $g(z)= 3z^2 $ B) $f(z) = 0$ and $g(z)= 0 $ C) $f(z) = 1$ and $g(z) = 0$ D) $f = 1$ and $g= z$ I didn't understand the question itself. Please someone help!! Thank you 
June 1st, 2017, 08:00 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
Can $2x + yf(z)$, $2y + xf(z)$ and $xyg(z)$ be the partial derivatives with respect to $x$, $y$ and $z$ respectively of some (scalar) function of $x$, $y$ and $z$?

June 1st, 2017, 04:11 PM  #3 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2 
Option A & B can be represented as a partial derivative of $x^2+xyz^3+y^2$ and $x^2+y^2$. Please correct me If wrong. 

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