# Del operator crossed with a scalar times a vector proof

• galactic
In summary, the proof for the identity \nabla\times\left(\phi\vec{V}\right)=(\phi\nabla)\times\vec{V}-\vec{V}\times(\nabla)\phi breaks down into three steps: 1) apply the product rule for differentiation, 2) reconstruct vectors from their components, and 3) apply the identity for the cross product \vec{A}\times(\vec{B}\times\vec{C})=(\vec{A}\cdot\vec{C})\vec{B}-(\vec{A}\cdot\vec{B})\vec{C}.
galactic
"Del" operator crossed with a scalar times a vector proof

## Homework Statement

Prove the following identity (we use the summation convention notation)

$$\bigtriangledown\times(\phi\vec{V})=(\phi \bigtriangledown)\times\vec{V}-\vec{V}\times(\bigtriangledown)\phi$$

## Homework Equations

equation for del, the gradient, curl..

## The Attempt at a Solution

im kind of confused on the first step...I broke it down into the following; however, levi civita symbols aren't my cup of tea and I get pretty confused about it...anyway here's what I did:

$$\bigtriangledown\times(\phi\vec{V})=(\epsilon_{ijk})\partial_i\vec{V}\phi\hat{x}_k$$

I don't know if this first step is right or if I decomposed the cross product right ?

Last edited by a moderator:

You should have the j:th component of V in your expression,
$$\nabla \times (\phi \vec{V}) = \epsilon_{ijk} \partial_i (\phi V_j) \hat{x}_k$$

The form you've written is completely equal to the one seen in math/physics books

$$\nabla\times \left(\phi\vec{V}\right) = \nabla\phi\times\vec{V} + \phi\nabla\times\vec{V}$$

Thanks for the replies, I'm just not sure what to do after what clamtrox said to do, the whole proofing business if pretty new to me :/

You need two more steps. Apply the product rule for differentiation and then once you obtain a sum of two terms, reconstruct vectors from their components.

## 1. What is the "Del operator" in this proof?

The Del operator, denoted by ∇, is a vector differential operator commonly used in vector calculus and physics. It represents the gradient, divergence, and curl operations on vector fields.

## 2. What does it mean to cross the Del operator with a scalar times a vector?

Crossing the Del operator (∇) with a scalar (α) times a vector (V) means performing the cross product of ∇ and αV, resulting in a new vector.

## 3. Why is this proof important in the field of science?

This proof is important because it helps in solving various mathematical problems in vector calculus and physics. It also has applications in fluid mechanics, electromagnetism, and other fields of science.

## 4. How is this proof derived?

This proof is derived using the properties of the Del operator and the cross product, as well as the definition of a scalar and vector. It involves algebraic manipulations and vector calculus techniques.

## 5. What are some real-life applications of this proof?

This proof has many practical applications, such as in electromagnetics for determining the electric and magnetic field intensities, in fluid dynamics for calculating the velocity and vorticity fields, and in differential geometry for studying manifolds and curves.

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