Q 67. - Calculus

Question

Prove that $\nabla f=0$ if $f$ is the constant function $f(x, y)=c$.

Step-by-Step Solution

Answer

Solve for $\nabla f=i\frac{\partial f}{\partial x}+j\frac{\partial f}{\partial y}$ to prove the above result.

Step 1: Given Information

It is given that $f(x, y)=c$.

Step 2: Simplification

Using $\nabla f=i\frac{\partial f}{\partial x}+j\frac{\partial f}{\partial y}$

$\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$

Hence

$\nabla f(x,y)=i\frac{\partial f}{\partial x}+j\frac{\partial f}{\partial y}$

$\nabla f(x,y)=i\left(0\right)+j\left(0\right)$

$\nabla f(x,y)=0$

Therefore $\nabla f(x,y)=0$