Question

In the usual notation, the value of $∆\nabla$ is equal to

• A. $∆-\nabla$
• B. $∆+\nabla$
• C. $\nabla -∆$
• D. None of these

Answer 1

The correct option is A

$∆-\nabla$

Explanation for the correct answer:

Del, or nabla, is an operator used in mathematics, in particular in vector calculus, as a vector differential operator, usually represented by the nabla symbol $\nabla$

$∆$ is known as the increment operator

Consider a function $f\left(x\right)$

Using the standard notation we can write

$\nabla f\left(x\right)=f\left(x\right)-f\left(x-h\right)$

$\Rightarrow$$∆\nabla f\left(x\right)=∆\left(f\left(x\right)-f\left(x-h\right)\right)$

$\Rightarrow$$∆\nabla f\left(x\right)=∆f\left(x\right)-∆f\left(x-h\right)$

$\Rightarrow$$∆\nabla f\left(x\right)=∆f\left(x\right)-\left(f\left(x\right)-f\left(x-h\right)\right)$ $\left[\because ∆f\left(x\right)=f(x+h)-f\left(x\right)\right]$

$\Rightarrow$$∆\nabla f\left(x\right)=∆f\left(x\right)-\nabla f\left(x\right)$

$\Rightarrow$ $∆\nabla =∆-\nabla$

Hence, option (A) is the correct answer.